THE
ÉTALE
THETA
FUNCTION
AND
ITS
FROBENIOID-THEORETIC
MANIFESTATIONS
Shinichi
Mochizuki
December
2008
We
develop
the
theory
of
the
tempered
anabelian
and
Frobenioid-theore-
tic
aspects
of
the
“étale
theta
function”,
i.e.,
the
Kummer
class
of
the
classical
formal
algebraic
theta
function
associated
to
a
Tate
curve
over
a
nonarchimedean
mixed-
characteristic
local
field.
In
particular,
we
consider
a
certain
natural
“environment”
for
the
study
of
the
étale
theta
function,
which
we
refer
to
as
a
“mono-theta
environ-
ment”
—
essentially
a
Kummer-theoretic
version
of
the
classical
theta
trivialization
—
and
show
that
this
mono-theta
environment
satisfies
certain
remarkable
rigidity
properties
involving
cyclotomes,
discreteness,
and
constant
multiples,
all
in
a
fash-
ion
that
is
compatible
with
the
topology
of
the
tempered
fundamental
group
and
the
extension
structure
of
the
associated
tempered
Frobenioid.
Contents:
§0.
Notations
and
Conventions
§1.
The
Tempered
Anabelian
Rigidity
of
the
Étale
Theta
Function
§2.
The
Theory
of
Theta
Environments
§3.
Tempered
Frobenioids
§4.
General
Bi-Kummer
Theory
§5.
The
Étale
Theta
Function
via
Tempered
Frobenioids
Introduction
The
fundamental
goal
of
the
present
paper
is
to
study
the
tempered
anabelian
[cf.
[André],
[Mzk14]]
and
Frobenioid-theoretic
[cf.
[Mzk17],
[Mzk18]]
aspects
of
the
theta
function
of
a
Tate
curve
over
a
nonarchimedean
mixed-characteristic
lo-
cal
field.
The
motivation
for
this
approach
to
the
theta
function
arises
from
the
long-term
goal
of
overcoming
various
obstacles
that
occur
when
one
attempts
to
apply
the
Hodge-Arakelov
theory
of
elliptic
curves
[cf.
[Mzk4],
[Mzk5];
[Mzk6],
[Mzk7],
[Mzk8],
[Mzk9],
[Mzk10]]
to
the
diophantine
geometry
of
elliptic
curves
over
number
fields.
That
is
to
say,
the
theory
of
the
present
paper
is
motivated
2000
Mathematical
Subject
Classification:
Primary
14H42;
Secondary
14H30.
Typeset
by
AMS-TEX
1
2
SHINICHI
MOCHIZUKI
by
the
expectation
that
these
obstacles
may
be
overcome
by
translating
the
[es-
sentially]
scheme-theoretic
formulation
of
Hodge-Arakelov
theory
into
the
language
of
the
geometry
of
categories
[e.g.,
the
“temperoids”
of
[Mzk14],
and
the
“Frobe-
nioids”
of
[Mzk17],
[Mzk18]].
In
certain
respects,
this
situation
is
reminiscent
of
the
well-known
classical
solution
to
the
problem
of
relating
the
dimension
of
the
first
cohomology
group
of
the
structure
sheaf
of
a
smooth
proper
variety
in
positive
characteristic
to
the
dimension
of
its
Picard
variety
—
a
problem
whose
solution
remained
elusive
until
the
foundations
of
the
algebraic
geometry
of
varieties
were
reformulated
in
the
language
of
schemes
[i.e.,
one
allows
for
the
possibility
of
nilpo-
tent
sections
of
the
structure
sheaf].
Since
Hodge-Arakelov
theory
centers
around
the
theory
of
the
theta
function
of
an
elliptic
curve
with
bad
multiplicative
reduction
[i.e.,
a
“Tate
curve”],
it
is
natural
to
attempt
to
begin
such
a
translation
by
concentrating
on
such
theta
functions
on
Tate
curves,
as
is
done
in
the
present
paper.
Indeed,
Hodge-Arakelov
theory
may
be
thought
of
as
a
sort
of
“canonical
analytic
continuation”
of
the
theory
of
theta
functions
on
Tate
curves
to
elliptic
curves
over
number
fields.
Here,
we
recall
that
although
classically,
the
arithmetic
theory
of
theta
functions
on
Tate
curves
is
developed
in
the
language
of
formal
schemes
in,
for
instance,
[Mumf]
[cf.
[Mumf],
pp.
306-307],
this
theory
only
addresses
the
“slope
zero
portion”
of
the
theory
—
i.e.,
the
portion
of
the
theory
that
involves
the
quotient
of
the
fundamental
group
of
the
generic
fiber
of
the
Tate
curve
that
extends
to
an
étale
covering
in
positive
characteristic.
From
this
point
of
view,
the
relation
of
the
theory
of
§1,
§2
of
the
present
paper
to
the
theory
of
[Mumf]
may
be
regarded
as
roughly
analogous
to
the
relation
of
the
theory
of
p-adic
uniformizations
of
hyperbolic
curves
developed
in
[Mzk1]
to
Mumford’s
theory
of
Schottky
uniformations
of
hyperbolic
curves
[cf.,
e.g.,
[Mzk1],
Introduction,
§0.1;
cf.
also
Remark
5.10.2
of
the
present
paper].
Frequently
in
classical
scheme-theoretic
constructions,
such
as
those
that
ap-
pear
in
the
scheme-theoretic
formulation
of
Hodge-Arakelov
theory,
there
is
a
ten-
dency
to
make
various
arbitrary
choices
in
situations
where,
a
priori,
some
sort
of
indeterminacy
exists,
without
providing
any
sort
of
intrinsic
justification
for
these
choices.
Typical
examples
of
such
choices
involve
the
choice
of
a
particular
rational
function
or
section
of
a
line
bundle
among
various
possibilities
related
by
a
constant
multiple,
or
the
choice
of
a
natural
identification
between
various
“cyclotomes”
[i.e.,
isomorphic
copies
of
the
module
of
N
-th
roots
of
unity,
for
N
≥
1
an
integer]
ap-
pearing
in
a
situation
[cf.,
e.g.,
[Mzk13],
Theorem
4.3,
for
an
example
of
a
crucial
rigidity
result
in
anabelian
geometry
concerning
this
sort
of
“choice”].
One
important
theme
of
the
present
paper
is
the
study
of
phenomena
that
involve
some
sort
of
“extraordinary
rigidity”
[cf.
Grothendieck’s
famous
use
of
this
expression
in
describing
his
anabelian
philosophy].
This
sort
of
“extraordinary
rigid-
ity”
may
be
thought
of
as
a
manifestation
of
the
very
strong
canonicality
present
in
the
theory
of
theta
functions
on
Tate
curves
that
allows
one,
in
Hodge-Arakelov
theory,
to
effect
a
“canonical
analytic
continuation”
of
this
theory
on
Tate
curves
to
elliptic
curves
over
global
number
fields.
The
various
examples
of
“extraordinary
rigidity”
that
appear
in
the
theory
of
the
étale
theta
function
may
be
thought
of
as
examples
of
intrinsic,
category-theoretic
“justifications”
for
the
arbitrary
choices
that
appear
in
classical
scheme-theoretic
discussions.
Such
category-theoretic
“jus-
THE
ÉTALE
THETA
FUNCTION
3
tifications”
depend
heavily
on
the
“proper
category-theoretic
formulation”
of
vari-
ous
scheme-theoretic
“venues”.
In
the
theory
of
the
present
paper,
one
central
such
category-theoretic
formulation
is
a
mathematical
structure
that
we
shall
refer
to
as
a
mono-theta
environment
[cf.
Definition
2.13,
(ii)].
Roughly
speaking:
The
mono-theta
environment
is
essentially
a
Kummer-theoretic
version
—
i.e.,
a
Galois-theoretic
version
obtained
by
extracting
various
N
-th
roots
[for
N
≥
1
an
integer]
—
of
the
theta
trivialization
that
appears
in
classical
formal
scheme-theoretic
discussions
of
the
theta
function
on
a
Tate
curve.
A
mono-theta
environment
may
be
thought
of
as
a
sort
of
common
core
for,
or
bridge
between,
the
[tempered]
étale-
and
Frobenioid-theoretic
approaches
to
the
étale
theta
function
[cf.
Remarks
2.18.2,
5.10.1,
5.10.2,
5.10.3].
We
are
now
ready
to
discuss
the
main
results
of
the
present
paper.
First,
we
remark
that
the
reader
who
is
only
interested
in
the
definition
and
basic
tempered
an-
abelian
properties
of
the
étale
theta
function
may
restrict
his/her
attention
to
the
theory
of
§1,
the
main
result
of
which
is
a
tempered
anabelian
con-
struction
of
the
étale
theta
function
[cf.
Theorems
1.6,
1.10].
On
the
other
hand,
the
main
results
of
the
bulk
of
the
paper,
which
concern
more
subtle
rigidity
properties
of
the
étale
theta
function
involving
associated
mono-theta
environments
and
tempered
Frobenioids,
may
be
summarized
as
follows:
A
mono-theta
environment
is
a
category/group-theoretic
invariant
of
the
tempered
étale
fundamental
group
[cf.
Corollary
2.18]
associated
to
[certain
coverings
of]
a
punctured
elliptic
curve
[satisfying
certain
properties]
over
a
nonarchimedean
mixed-characteristic
local
field,
on
the
one
hand,
and
of
a
certain
tempered
Frobenioid
[cf.
Theorem
5.10,
(iii)]
associated
to
such
a
curve,
on
the
other.
Moreover,
a
mono-theta
environment
satisfies
the
following
rigidity
properties:
(a)
cyclotomic
rigidity
[cf.
Corollary
2.19,
(i);
Remark
2.19.4];
(b)
discrete
rigidity
[cf.
Corollary
2.19,
(ii);
Remarks
2.16.1,
2.19.4];
(c)
constant
multiple
rigidity
[cf.
Corollary
2.19,
(iii);
Remarks
5.12.3,
5.12.5]
—
all
in
a
fashion
that
is
compatible
with
the
topology
of
the
tempered
fundamental
group
as
well
as
with
the
extension
structure
of
the
tem-
pered
Frobenioid
[cf.
Corollary
5.12
and
the
discussion
of
the
following
remarks].
In
particular,
the
phenomenon
of
“cyclotomic
rigidity”
gives
a
“category-theoretic”
explanation
for
the
special
role
played
by
the
first
power
[i.e.,
as
opposed
to
the
4
SHINICHI
MOCHIZUKI
M
-th
power,
for
M
>
1
an
integer]
of
[the
l-th
root,
when
one
works
with
l-torsion
points,
for
l
≥
1
an
odd
integer,
of]
the
theta
function
[cf.
Remarks
2.19.2,
2.19.3,
5.10.3,
5.12.5]
—
a
phenomenon
which
may
also
be
seen
in
“scheme-theoretic”
Hodge-Arakelov
theory
[a
theory
that
also
effectively
involves
[sections
of]
the
first
power
of
some
ample
line
bundle,
and
which
does
not
generalize
in
any
evident
way
to
arbitrary
powers
of
this
ample
line
bundle].
One
further
interesting
aspect
of
cyclotomic
rigidity
in
the
theory
of
the
present
paper
is
that
it
is
obtained
essentially
as
the
result
of
a
certain
computation
in-
volving
commutators
[cf.
Remark
2.19.2].
Put
another
way,
one
may
think
of
this
cyclotomic
rigidity
as
a
sort
of
consequence
of
the
nonabelian
structure
of
[what
es-
sentially
amounts,
from
a
more
classical
point
of
view,
to]
the
“theta-group”.
This
application
of
the
structure
of
the
theta-group
differs
substantially
from
the
way
in
which
the
structure
of
the
theta-group
is
used
in
more
classical
treatments
of
the
theory
of
theta
functions
—
namely,
to
show
that
certain
representations,
such
as
those
arising
from
theta
functions,
are
irreducible.
One
way
to
understand
this
difference
is
as
follows.
Whereas
classical
treatments
that
center
around
such
ir-
reducibility
results
only
treat
certain
l-dimensional
subspaces
of
the
l
2
-dimensional
space
of
set-theoretic
functions
on
the
l-torsion
points
of
an
elliptic
curve,
Hodge-
Arakelov
theory
is
concerned
with
understanding
[“modulo”
the
l-dimensional
sub-
space
which
has
already
been
well
understood
in
the
classical
theory!]
the
entire
l
2
-dimensional
function
space
that
appears
[cf.
[Mzk4],
§1.1].
Put
another
way,
the
l-dimensional
subspace
which
forms
the
principal
topic
of
the
classical
theory
may
be
thought
of
as
corresponding
to
the
space
of
holomorphic
functions
[cf.
[Mzk4],
§1.4.2];
by
contrast,
Hodge-Arakelov
theory
—
cf.,
e.g.,
the
arithmetic
Kodaira-
Spencer
morphism
of
[Mzk4],
§1.4
—
is
concerned
with
understanding
deforma-
tions
of
the
holomorphic
structure,
from
an
arithmetic
point
of
view.
Moreover,
from
the
point
of
view
of
considering
such
deformations
of
holomorphic
structure,
it
is
convenient,
and,
indeed,
more
efficient,
to
work
“modulo
variations
contained
within
the
subspace
corresponding
to
the
holomorphic
functions”
—
which,
at
any
rate,
may
be
treated,
as
a
consequence
of
the
classical
theory
of
irreducible
repre-
sentations
of
the
theta-group,
as
a
single
“irreducible
unit”!
This
is
precisely
the
point
of
view
of
the
“Lagrangian
approach”
of
[Mzk5],
§3,
an
approach
which
allows
one
to
work
“modulo
the
l-dimensional
subspace
of
the
classical
theory”,
by
apply-
ing
various
isogenies
that
allow
one
to
replace
the
“entire
l
2
-dimensional
function
space”
associated
to
the
original
elliptic
curve
by
an
l-dimensional
function
space
that
is
suited
to
studying
“arithmetic
deformations
of
holomorphic
structure”
in
the
style
of
Hodge-Arakelov
theory.
In
the
present
paper,
these
isogenies
of
the
“La-
grangian
approach
to
Hodge-Arakelov
theory”
correspond
to
the
various
coverings
that
appear
in
the
discussion
at
the
beginning
of
§2;
the
resulting
“l-dimensional
function
space”
then
corresponds,
in
the
theory
of
the
present
paper,
to
the
space
of
functions
on
the
labels
that
appear
in
Corollary
2.9.
Here,
it
is
important
to
note
that
although
the
above
three
rigidity
proper-
ties
may
be
stated
and
understood
to
a
certain
extent
without
reference
to
the
Frobenioid-theoretic
portion
of
the
theory
[cf.
§2],
certain
aspects
of
the
interde-
pendence
of
these
rigidity
properties,
as
well
as
the
meaning
of
establishing
these
rigidity
properties
under
the
condition
of
compatibility
with
the
topology
of
the
THE
ÉTALE
THETA
FUNCTION
5
tempered
fundamental
group
as
well
as
with
the
extension
structure
of
the
tempered
Frobenioid,
may
only
be
understood
in
the
context
of
the
theory
of
Frobenioids
[cf.
Corollary
5.12
and
the
discussion
of
the
following
remarks].
Indeed,
one
important
theme
of
the
theory
of
the
present
paper
—
which
may,
roughly,
be
summarized
as
the
idea
that
sometimes
“less
(respectively,
more)
data
yields
more
(respectively,
less)
information”
[cf.
Remark
5.12.9]
—
is
precisely
the
study
of
the
rather
intricate
way
in
which
these
various
rigidity/compatibility
properties
are
related
to
one
another.
Typically
speaking,
the
reason
for
this
[at
first
glance
somewhat
paradoxical]
phenomenon
is
that
“more
data”
means
more
complicated
systems
made
up
of
various
components,
hence
obligates
one
to
keep
track
of
the
various
indeterminacies
that
arise
in
re-
lating
[i.e.,
without
resorting
to
the
use
of
“arbitrary
choices”]
these
components
to
one
another.
If
these
indeterminacies
are
sufficiently
severe,
then
they
may
have
the
effect
of
obliterating
certain
structures
that
one
is
interested
in.
By
contrast,
if
certain
portions
of
such
a
system
are
redundant,
i.e.,
in
fact
uniquely
and
rigidly
—
i.e.,
“canonically”
—
determined
by
more
fundamental
portions
of
the
system,
then
one
need
not
contend
with
the
indeterminacies
that
arise
from
relating
the
redundant
components;
this
yields
a
greater
chance
that
the
structures
of
interest
are
not
obliterated
by
the
intrinsic
indeterminacies
of
the
system.
One
way
to
appreciate
the
“tension”
that
exists
between
the
various
rigidity
properties
satisfied
by
the
mono-theta
environment
is
by
comparing
the
theory
of
the
mono-theta
environment
to
that
of
the
bi-theta
environment
[cf.
Definition
2.13,
(iii)].
The
bi-theta
environment
is
essentially
a
Kummer-theoretic
version
of
the
pair
of
sections
—
corresponding
to
the
“numerator”
and
“denominator”
of
the
theta
function
—
of
a
certain
ample
line
bundle
on
a
Tate
curve.
One
of
these
two
sections
is
the
theta
trivialization
that
appears
in
the
mono-theta
environment;
the
other
of
these
two
sections
is
the
“algebraic
section”
that
arises
tautologically
from
the
original
definition
of
the
ample
line
bundle.
The
bi-theta
environment
satisfies
cyclotomic
rigidity
and
constant
multiple
rigidity
properties
for
somewhat
more
evident
reasons
than
the
mono-theta
envi-
ronment.
In
the
case
of
constant
multiple
rigidity,
this
arises
partly
from
the
fact
that
the
bi-theta
environment
involves
working,
in
essence,
with
a
ratio
[i.e.,
in
the
form
of
a
“pair”]
of
sections,
hence
is
immune
to
the
operation
of
multiplying
both
sections
by
a
constant
[cf.
Remarks
5.10.4;
5.12.7,
(ii)].
On
the
other
hand,
the
bi-theta
environment
fails
to
satisfy
the
discrete
rigidity
property
satisfied
by
the
mono-theta
environment
[cf.
Corollary
2.16;
Remark
2.16.1].
By
contrast,
the
mono-theta
environment
satisfies
all
three
rigidity
properties,
despite
the
fact
that
cyclotomic
rigidity
[cf.
the
proof
of
Corollary
2.19,
(i);
Remark
2.19.2]
and
constant
multiple
rigidity
[cf.
Remarks
5.12.3,
5.12.5]
are
somewhat
more
subtle.
Here,
it
is
interesting
to
note
that
the
issue
of
discrete
rigidity
for
the
mono-
theta
and
bi-theta
environments
revolves,
in
essence,
around
the
fact
that
Z/Z
=
0,
i.e.,
the
gap
between
Z
and
its
profinite
completion
Z
—
cf.
Remark
2.16.1.
On
the
other
hand,
the
subtlety
of
constant
multiple
rigidity
for
mono-theta
environ-
ments
revolves,
in
essence,
around
the
nontriviality
of
the
extension
structure
of
6
SHINICHI
MOCHIZUKI
a
Frobenoid
[i.e.,
as
an
“extension
by
line
bundles
of
the
base
category”
—
cf.
Remarks
5.10.2,
5.12.3,
5.12.5,
5.12.7].
Put
another
way:
The
mono-theta
environment
may
be
thought
of
as
a
sort
of
translation
apparatus
that
serves
to
translate
the
“global
arithmetic
gap
between
[cf.
Remark
2.16.2
for
more
on
the
relation
of
this
portion
Z
and
Z”
of
the
theory
to
global
arithmetic
bases]
into
the
“nontriviality
of
the
local
geometric
extension”
constituted
by
the
extension
structure
of
the
tempered
Frobenioid
under
consideration.
and
This
sort
of
relationship
between
the
global
arithmetic
gap
between
Z
and
Z
the
theory
of
theta
functions
is
reminiscent
of
the
point
of
view
that
theta
functions
are
related
to
“splittings
of
the
natural
surjection
Z
Z/N
Z”,
a
point
of
view
that
arises
in
Hodge-Arakelov
theory
[cf.
[Mzk4],
§1.3.3].
The
contents
of
the
present
paper
are
organized
as
follows.
In
§1,
we
discuss
the
purely
tempered
étale-theoretic
anabelian
aspects
of
the
theta
function
and
show,
in
particular,
that
the
“étale
theta
function”
—
i.e.,
the
Kummer
class
of
the
usual
formal
algebraic
theta
function
—
is
preserved
by
isomorphisms
of
the
tempered
fundamental
group
[cf.
Theorems
1.6;
1.10].
In
§2,
after
studying
various
coverings
and
quotient
coverings
of
a
punctured
elliptic
curve,
we
introduce
the
notions
of
a
mono-theta
environment
and
a
bi-theta
environment
[cf.
Definition
2.13]
and
study
the
“group-theoretic
constructibility”
and
rigidity
properties
of
these
notions
[cf.
Corollaries
2.18,
2.19].
In
§3,
we
define
the
“tempered
Frobenioids”
in
which
we
shall
develop
the
Frobenioid-theoretic
approach
to
the
étale
theta
function;
in
particular,
we
show
that
these
tempered
Frobenioids
satisfy
various
nice
properties
which
allow
one
to
apply
the
extensive
theory
of
[Mzk17],
[Mzk18]
[cf.
Theorem
3.7;
Corollary
3.8].
In
§4,
we
develop
“bi-Kummer
theory”
—
i.e.,
a
sort
of
generalization
of
the
“Kummer
class
associated
to
a
rational
function”
to
the
“bi-Kummer
data”
associated
to
a
pair
of
sections
[corresponding
to
the
numerator
and
denominator
of
a
rational
function]
of
a
line
bundle
—
in
a
category-theoretic
fashion
[cf.
Theorem
4.4]
for
fairly
general
tempered
Frobenioids.
Finally,
in
§5,
we
specialize
the
theory
of
§3,
§4
to
the
case
of
the
étale
theta
function,
as
discussed
in
§1.
In
particular,
we
observe
that
a
mono-theta
environment
may
also
be
regarded
as
a
mathematical
structure
naturally
associated
to
a
certain
tempered
Frobenioid
[cf.
Theorem
5.10,
(iii)].
Also,
we
discuss
certain
aspects
of
the
constant
multiple
rigidity
[as
well
as,
to
a
lesser
extent,
of
the
cyclotomic
and
discrete
rigidity]
of
a
mono-theta
environment
that
may
only
be
understood
in
the
context
of
the
Frobenioid-theoretic
approach
to
the
étale
theta
function
[cf.
Corollary
5.12
and
the
discussion
of
the
following
remarks].
Acknowledgements:
I
would
like
to
thank
Akio
Tamagawa,
Makoto
Matsumoto,
Minhyong
Kim,
Seidai
Yasuda,
Kazuhiro
Fujiwara,
and
Fumiharu
Kato
for
many
helpful
comments
concerning
the
material
presented
in
this
paper.
THE
ÉTALE
THETA
FUNCTION
7
Section
0:
Notations
and
Conventions
In
addition
to
the
“Notations
and
Conventions”
of
[Mzk17],
§0,
we
shall
employ
the
following
“Notations
and
Conventions”
in
the
present
paper:
Monoids:
We
shall
denote
by
N
≥1
the
multiplicative
monoid
of
[rational]
integers
≥
1
[cf.
[Mzk17],
§0].
Let
Q
be
a
commutative
monoid
[with
unity];
P
⊆
Q
a
submonoid.
If
Q
is
integral
[so
Q
embeds
into
its
groupification
Q
gp
;
we
have
a
natural
inclusion
P
gp
→
Q
gp
],
then
we
shall
refer
to
the
submonoid
P
gp
Q
(⊆
Q
gp
)
of
Q
as
the
group-saturation
of
P
in
Q;
if
P
is
equal
to
its
group-saturation
in
Q,
then
we
shall
say
that
P
is
group-saturated
in
Q.
If
Q
is
torsion-free
[so
Q
embeds
into
its
perfection
Q
pf
;
we
have
a
natural
inclusion
P
pf
→
Q
pf
],
then
we
shall
refer
to
the
submonoid
P
pf
Q
(⊆
Q
pf
)
of
Q
as
the
perf-saturation
of
P
in
Q;
if
P
is
equal
to
its
perf-saturation
in
Q,
then
we
shall
say
that
P
is
perf-saturated
in
Q.
Topological
Groups:
Let
Π
be
a
topological
group.
Then
let
us
write
B
temp
(Π)
for
the
category
whose
objects
are
countable
[i.e.,
of
cardinality
≤
the
cardinality
of
the
set
of
natural
numbers],
discrete
sets
equipped
with
a
continuous
Π-action
and
whose
morphisms
are
morphisms
of
Π-sets
[cf.
[Mzk14],
§3].
If
Π
may
be
written
as
an
inverse
limit
of
an
inverse
system
of
surjections
of
countable
discrete
topological
groups,
then
we
shall
say
that
Π
is
tempered
[cf.
[Mzk14],
Definition
3.1,
(i)].
We
shall
refer
to
a
normal
open
subgroup
H
⊆
Π
such
that
the
quotient
group
Π/H
is
free
as
co-free.
We
shall
refer
to
a
co-free
subgroup
H
⊆
Π
as
minimal
if
every
co-free
subgroup
of
Π
contains
H.
Thus,
a
minimal
co-free
subgroup
of
Π
is
necessarily
unique
and
characteristic.
Categories:
We
shall
refer
to
an
isomorphic
copy
of
some
object
as
an
isomorph
of
the
object.
8
SHINICHI
MOCHIZUKI
Let
C
be
a
category;
A
∈
Ob(C).
Then
we
shall
write
C
A
for
the
category
whose
objects
are
morphisms
B
→
A
of
C
and
whose
morphisms
[from
an
object
B
1
→
A
to
an
object
B
2
→
A]
are
A-morphisms
B
1
→
B
2
in
C
[cf.
[Mzk17],
§0]
and
C[A]
⊆
C
for
the
full
subcategory
of
C
determined
by
the
objects
of
C
that
admit
a
morphism
to
A.
Given
two
arrows
f
i
:
A
i
→
B
i
(where
i
=
1,
2)
in
C,
we
shall
refer
to
a
commutative
diagram
∼
A
1
→
A
2
⏐
⏐
⏐
f
⏐
f
1
2
B
1
∼
→
B
2
—
where
the
horizontal
arrows
are
isomorphisms
in
C
—
as
an
abstract
equivalence
from
f
1
to
f
2
.
If
there
exists
an
abstract
equivalence
from
f
1
to
f
2
,
then
we
shall
say
that
f
1
,
f
2
are
abstractly
equivalent.
Let
Φ
:
C
→
D
be
a
faithful
functor
between
categories
C,
D.
Then
we
shall
∼
say
that
Φ
is
isomorphism-full
if
every
isomorphism
Φ(A)
→
Φ(B)
of
D,
where
∼
A,
B
∈
Ob(C),
arises
by
applying
Φ
to
an
isomorphism
A
→
B
of
C.
Suppose
that
Φ
is
isomorphism-full.
Then
observe
that
the
objects
of
D
that
are
isomorphic
to
objects
in
the
image
of
Φ,
together
with
the
morphisms
of
D
that
are
abstractly
equivalent
to
morphisms
in
the
image
of
Φ,
form
a
subcategory
C
⊆
D
such
that
∼
Φ
induces
an
equivalence
of
categories
C
→
C
.
We
shall
refer
to
this
subcategory
C
⊆
D
as
the
essential
image
of
Φ.
[Thus,
this
terminology
is
consistent
with
the
usual
terminology
of
“essential
image”
in
the
case
where
Φ
is
fully
faithful.]
Curves:
We
refer
to
[Mzk14],
§0,
for
generalities
concerning
[families
of
]
hyperbolic
curves,
smooth
log
curves,
stable
log
curves,
divisors
of
cusps,
and
divisors
of
marked
points.
If
C
log
→
S
log
is
a
stable
log
curve,
and,
moreover,
S
is
the
spectrum
of
a
field,
then
we
shall
say
that
C
log
is
split
if
each
of
the
irreducible
components
and
nodes
of
C
is
geometrically
irreducible
over
S.
A
morphism
of
log
stacks
C
log
→
S
log
for
which
there
exists
an
étale
surjection
S
1
→
S,
where
S
1
is
a
scheme,
such
that
def
C
1
log
=
C
log
×
S
S
1
may
be
obtained
as
the
result
of
forming
the
quotient
[in
the
sense
def
of
log
stacks!]
of
a
stable
(respectively,
smooth)
log
curve
C
2
log
→
S
1
log
=
S
log
×
S
S
1
by
the
action
of
a
finite
group
of
automorphisms
of
C
2
log
over
S
1
log
which
acts
freely
on
a
dense
open
subset
of
every
fiber
of
C
2
→
S
1
will
be
referred
to
as
a
stable
log
THE
ÉTALE
THETA
FUNCTION
9
orbicurve
(respectively,
smooth
log
orbicurve)
over
S
log
.
Thus,
the
divisor
of
cusps
of
C
2
log
determines
a
divisor
of
cusps
of
C
1
log
,
C
log
.
Here,
if
C
2
log
→
S
1
log
is
of
type
(1,
1),
and
the
finite
group
of
automorphisms
is
given
by
the
action
of
“±1”
[i.e.,
relative
to
the
group
structure
of
the
underlying
elliptic
curve
of
C
2
log
→
S
1
log
],
then
the
resulting
stable
log
orbicurve
will
be
referred
to
as
being
of
type
(1,
1)
±
.
If
S
log
is
the
spectrum
of
a
field,
equipped
with
the
trivial
log
structure,
then
a
hyperbolic
orbicurve
X
→
S
is
defined
to
be
the
algebraic
[log]
stack
[with
trivial
log
structure]
obtained
by
removing
the
divisor
of
cusps
from
some
smooth
log
orbicurve
C
log
→
S
log
over
S
log
.
If
X
(respectively,
Y
)
is
a
hyperbolic
orbicurve
over
a
field
K
(respectively,
L),
then
we
shall
say
that
X
is
isogenous
to
Y
if
there
exists
a
hyperbolic
curve
Z
over
a
field
M
together
with
finite
étale
morphisms
Z
→
X,
Z
→
Y
.
Note
that
in
this
situation,
the
morphisms
Z
→
X,
Z
→
Y
induce
finite
separable
inclusions
of
fields
K
→
M
,
L
→
M
.
[Indeed,
this
follows
immediately
×
)
such
that
G
{0}
from
the
easily
verified
fact
that
every
subgroup
G
⊆
Γ(Z,
O
Z
determines
a
field
is
necessarily
contained
in
M
×
.]
10
SHINICHI
MOCHIZUKI
Section
1:
The
Tempered
Anabelian
Rigidity
of
the
Étale
Theta
Function
In
this
§,
we
construct
a
certain
cohomology
class
in
the
[continuous]
group
cohomology
of
the
tempered
fundamental
group
of
a
once-punctured
elliptic
curve
which
may
be
regarded
as
a
sort
of
“tempered
analytic”
representation
of
the
theta
function.
We
then
discuss
various
properties
of
this
“étale
theta
function”.
In
particular,
we
apply
the
theory
of
[Mzk14],
§6,
to
show
that
it
is,
up
to
certain
rela-
tively
mild
indeterminacies,
preserved
by
arbitrary
automorphisms
of
the
tempered
fundamental
group
[cf.
Theorems
1.6,
1.10].
Let
K
be
a
finite
extension
of
Q
p
,
with
ring
of
integers
O
K
;
K
an
algebraic
clo-
sure
of
K;
S
the
formal
scheme
determined
by
the
p-adic
completion
of
Spec(O
K
);
S
log
the
formal
log
scheme
obtained
by
equipping
S
with
the
log
structure
deter-
mined
by
the
unique
closed
point
of
Spec(O
K
);
X
log
a
stable
log
curve
over
S
log
of
type
(1,
1).
Also,
we
assume
that
the
special
fiber
of
X
is
singular
and
split
[cf.
§0],
and
that
the
generic
fiber
of
the
algebrization
of
X
log
is
a
smooth
log
curve.
Write
def
X
log
=
X
log
×
O
K
K
for
the
ringed
space
with
log
structure
obtained
by
tensoring
the
structure
sheaf
of
X
over
O
K
with
K.
In
the
following
discussion,
we
shall
often
[by
abuse
of
notation]
use
the
notation
X
log
also
to
denote
the
generic
fiber
of
the
algebrization
of
X
log
.
[Here,
the
reader
should
note
that
these
notational
conven-
tions
differ
somewhat
from
notational
conventions
typically
employed
in
discussions
of
rigid-analytic
geometry.]
Let
us
write
Π
tp
X
for
the
tempered
fundamental
group
associated
to
X
log
[cf.
[André],
§4;
the
group
log
)”
of
[Mzk14],
Examples
3.10,
5.6],
with
respect
to
some
basepoint.
The
“π
1
temp
(X
K
issue
of
how
our
constructions
are
affected
when
the
basepoint
varies
will
be
studied
in
the
present
paper
by
considering
to
what
extent
these
constructions
are
preserved
by
inner
or,
more
generally,
arbitrary
isomorphisms
between
fundamental
groups
[cf.
Remark
1.6.2
below].
Here,
despite
the
fact
that
the
[tempered]
fundamental
group
in
question
is
best
thought
of
not
as
“the
fundamental
group
of
X”
but
rather
tp
as
“the
fundamental
group
of
X
log
”,
we
use
the
notation
“Π
tp
X
”
rather
than
“Π
X
log
”
in
order
to
minimize
the
number
of
subscripts
and
superscripts
that
appear
in
the
notation
[cf.
the
discussion
to
follow
in
the
remainder
of
the
present
paper!];
thus,
the
reader
should
think
of
the
underlined
notation
“Π
tp
(−)
”
as
an
abbreviation
for
the
“logarithmic
tempered
fundamental
group
of
the
scheme
(−),
equipped
with
the
log
structure
currently
under
consideration”,
i.e.,
an
abbreviation
for
“π
1
temp
((−)
log
)”.
tp
Denote
by
Δ
tp
X
⊆
Π
X
the
“geometric
tempered
fundamental
group”.
Thus,
we
have
a
natural
exact
sequence
tp
1
→
Δ
tp
X
→
Π
X
→
G
K
→
1
def
—
where
G
K
=
Gal(K/K).
THE
ÉTALE
THETA
FUNCTION
11
Since
the
special
fiber
of
X
is
split,
it
follows
that
the
universal
graph-covering
of
the
dual
graph
of
this
special
fiber
determines
[up
to
composition
with
an
element
of
Aut(Z)
=
{±1}]
a
natural
surjection
Π
tp
X
Z
whose
kernel,
which
we
denote
by
Π
tp
Y
,
determines
an
infinite
étale
covering
Y
log
→
X
log
—
i.e.,
Y
log
is
a
p-adic
formal
scheme
equipped
with
a
log
structure;
the
special
fiber
of
Y
is
an
infinite
chain
of
copies
of
the
projective
line,
joined
at
0
and
∞;
def
def
write
Y
log
=
Y
log
×
O
K
K;
Z
=
Gal(Y
/X)
(
∼
=
Z).
def
def
tp
∧
∧
Write
Π
X
=
(Π
tp
[where
the
“∧”
denotes
the
profinite
X
)
;
Δ
X
=
(Δ
X
)
completion].
Then
we
have
a
natural
exact
sequence
1
→
Z(1)
→
Δ
ell
X
→
Z
→
1
def
ab
—
where
we
write
Δ
ell
X
=
Δ
X
=
Δ
X
/[Δ
X
,
Δ
X
]
for
the
abelianization
of
Δ
X
.
Since
Δ
X
is
a
profinite
free
group
on
2
generators,
we
also
have
a
natural
exact
sequence
Θ
ell
∼
1
→
∧
2
Δ
ell
X
(
=
Z(1))
→
Δ
X
→
Δ
X
→
1
def
ell
2
—
where
we
write
Δ
Θ
X
=
Δ
X
/[Δ
X
,
[Δ
X
,
Δ
X
]].
Let
us
denote
the
image
of
∧
Δ
X
Θ
∼
in
Δ
Θ
X
by
(
Z(1)
=)
Δ
Θ
⊆
Δ
X
.
Similarly,
we
have
natural
exact
sequences
ell
→
Z
→
1
1
→
Z(1)
→
(Δ
tp
X
)
tp
ell
Θ
→
1
1
→
Δ
Θ
→
(Δ
tp
X
)
→
(Δ
X
)
—
where
we
write
tp
Θ
tp
ell
Δ
tp
X
(Δ
X
)
(Δ
X
)
ell
for
the
quotients
induced
by
the
quotients
Δ
X
Δ
Θ
X
Δ
X
.
Also,
we
shall
write
tp
Θ
tp
ell
Π
tp
X
(Π
X
)
(Π
X
)
tp
Θ
for
the
quotients
whose
kernels
are
the
kernels
of
the
quotients
Δ
tp
X
(Δ
X
)
tp
ell
(Δ
X
)
and
tp
Θ
tp
ell
Π
tp
Y
(Π
Y
)
(Π
Y
)
;
tp
Θ
tp
ell
Δ
tp
Y
(Δ
Y
)
(Δ
Y
)
tp
tp
tp
for
the
quotients
of
Π
tp
Y
,
Δ
Y
induced
by
the
quotients
of
Π
X
,
Δ
X
with
similar
ell
∼
superscripts.
Thus,
(Δ
tp
=
Z(1);
we
have
a
natural
exact
sequence
of
abelian
Y
)
tp
ell
Θ
profinite
groups
1
→
Δ
Θ
→
(Δ
tp
→
1.
Y
)
→
(Δ
Y
)
12
SHINICHI
MOCHIZUKI
Next,
let
us
write
q
X
∈
O
K
for
the
q-parameter
of
the
underlying
elliptic
curve
of
X
log
.
If
N
≥
1
is
an
integer,
set
def
1/N
K
N
=
K(ζ
N
,
q
X
)
⊆
K
—
where
ζ
N
is
a
primitive
N
-th
root
of
unity.
Then
any
decomposition
group
of
tp
ell
ell
a
cusp
of
Y
log
determines,
up
to
conjugation
by
(Δ
tp
Y
)
,
a
section
G
K
→
(Π
Y
)
ell
of
the
natural
surjection
(Π
tp
G
K
whose
restriction
to
the
open
subgroup
Y
)
G
K
N
⊆
G
K
determines
an
open
immersion
tp
ell
ell
G
K
N
→
(Π
tp
Y
)
/N
·
(Δ
Y
)
the
image
of
which
is
stabilized
by
the
conjugation
action
of
Π
tp
X
.
[Indeed,
this
tp
ell
tp
ell
follows
from
the
fact
that
G
K
N
acts
trivially
on
(Δ
X
)
/N
·
(Δ
Y
)
.]
Thus,
this
image
determines
a
Galois
covering
Y
N
→
Y
such
that
the
resulting
surjection
Π
tp
Y
Gal(Y
N
/Y
),
whose
kernel
we
denote
by
tp
ell
Π
Y
N
,
induces
a
natural
exact
sequence
1
→
(Δ
tp
⊗
Z/N
Z
→
Gal(Y
N
/Y
)
→
Y
)
Gal(K
N
/K)
→
1.
Also,
we
shall
write
tp
Θ
tp
ell
Π
tp
Y
N
(Π
Y
N
)
(Π
Y
N
)
;
tp
Θ
tp
ell
Δ
tp
Y
N
(Δ
Y
N
)
(Δ
Y
N
)
tp
tp
tp
for
the
quotients
of
Π
tp
Y
N
,
Δ
Y
N
induced
by
the
quotients
of
Π
Y
,
Δ
Y
with
similar
superscripts
and
Y
N
log
for
the
object
obtained
by
equipping
Y
N
with
the
log
structure
determined
by
the
K
N
-valued
points
of
Y
N
lying
over
the
cusps
of
Y
.
Set
Y
N
→
Y
equal
to
the
normalization
of
Y
in
Y
N
.
One
verifies
easily
that
the
special
fiber
of
Y
N
is
an
infinite
chain
of
copies
of
the
projective
line,
joined
at
0
and
∞;
each
of
these
points
“0”
and
“∞”
is
a
node
on
Y
N
;
each
projective
line
in
this
chain
maps
to
a
projective
line
in
the
special
fiber
of
Y
by
the
“N
-th
power
map”
on
the
copy
of
“G
m
”
obtained
by
removing
the
nodes;
if
we
choose
some
irreducible
component
of
the
special
fiber
of
Y
as
a
“basepoint”,
then
the
natural
action
of
Z
on
Y
allows
one
to
think
of
the
projective
lines
in
the
special
fiber
of
Y
as
being
labeled
by
elements
of
Z.
In
particular,
it
follows
immediately
that
the
isomorphism
class
of
a
line
bundle
on
Y
N
is
completely
determined
by
the
degree
of
the
restriction
of
the
line
bundle
to
each
of
these
copies
of
the
projective
line.
That
is
to
say,
these
degrees
determine
an
isomorphism
∼
Pic(Y
N
)
→
Z
Z
THE
ÉTALE
THETA
FUNCTION
13
—
where
Z
Z
denotes
the
module
of
functions
Z
→
Z;
the
additive
structure
on
this
module
is
induced
by
the
additive
structure
on
the
codomain
“Z”.
Write
L
N
for
the
line
bundle
on
Y
N
determined
by
the
constant
function
Z
→
Z
whose
value
is
1.
Also,
we
observe
that
it
follows
immediately
from
the
above
explicit
description
of
the
special
fiber
of
Y
N
that
Γ(Y
N
,
O
Y
N
)
=
O
K
N
.
Next,
write
def
J
N
=
K
N
(a
1/N
)
a∈K
N
⊆
K
×
is
topologically
finitely
generated]
J
N
is
a
finite
—
where
we
note
that
[since
K
N
Galois
extension
of
K
N
.
Observe,
moreover,
that
we
have
an
exact
sequence
tp
tp
1
→
Δ
Θ
⊗
Z/N
Z
(
∼
=
Z/N
Z(1))
→
(Π
Y
N
)
Θ
/N
·
(Δ
Y
)
Θ
→
G
K
N
→
1
[cf.
the
construction
of
Y
N
].
Since
any
two
splittings
of
this
exact
sequence
differ
by
a
cohomology
class
∈
H
1
(G
K
N
,
Z/N
Z(1)),
it
follows
[by
the
definition
of
J
N
]
that
all
splittings
of
this
exact
sequence
determine
the
same
splitting
over
G
J
N
.
Thus,
the
image
of
the
resulting
open
immersion
tp
Θ
Θ
G
J
N
→
(Π
tp
Y
N
)
/N
·
(Δ
Y
)
is
stabilized
by
the
conjugation
action
of
Π
tp
X
,
hence
determines
a
Galois
covering
Z
N
→
Y
N
such
that
the
resulting
surjection
Π
tp
Y
N
Gal(Z
N
/Y
N
),
whose
kernel
we
denote
tp
by
Π
Z
N
,
induces
a
natural
exact
sequence
1
→
Δ
Θ
⊗
Z/N
Z
→
Gal(Z
N
/Y
N
)
→
Gal(J
N
/K
N
)
→
1.
Also,
we
shall
write
tp
Θ
tp
ell
Π
tp
Z
N
(Π
Z
N
)
(Π
Z
N
)
;
tp
Θ
tp
ell
Δ
tp
Z
N
(Δ
Z
N
)
(Δ
Z
N
)
tp
tp
tp
for
the
quotients
of
Π
tp
Z
N
,
Δ
Z
N
induced
by
the
quotients
of
Π
Y
N
,
Δ
Y
N
with
similar
superscripts
and
log
Z
N
for
the
object
obtained
by
equipping
Z
N
with
the
log
structure
determined
by
the
[manifestly]
J
N
-valued
points
of
Z
N
lying
over
the
cusps
of
Y
.
Set
Z
N
→
Y
N
equal
to
the
normalization
of
Y
in
Z
N
.
Since
Y
is
“generically
of
characteristic
zero”
[i.e.,
Y
is
of
characteristic
zero],
it
follows
that
Z
N
is
finite
over
Y.
Next,
let
us
observe
that
there
exists
a
section
s
1
∈
Γ(Y
=
Y
1
,
L
1
)
14
SHINICHI
MOCHIZUKI
×
—
well-defined
up
to
an
O
K
-multiple
—
whose
zero
locus
on
Y
is
precisely
the
divisor
of
cusps
of
Y.
Also,
let
us
fix
an
isomorphism
of
L
⊗N
with
L
1
|
Y
N
,
which
N
we
use
to
identify
these
two
bundles.
Note
that
there
is
a
natural
action
of
Gal(Y
/X)
on
L
1
which
is
uniquely
determined
by
the
condition
that
it
preserve
s
1
.
Thus,
we
obtain
a
natural
action
of
Gal(Y
N
/X)
on
L
1
|
Y
N
.
Proposition
1.1.
(Theta
Action
of
the
Tempered
Fundamental
Group)
(i)
The
section
s
1
|
Y
N
∈
Γ(Y
N
,
L
1
|
Y
N
∼
=
L
⊗N
N
)
admits
an
N
-th
root
s
N
∈
Γ(Z
N
,
L
N
|
Z
N
)
over
Z
N
.
In
particular,
if
we
denote
associated
“geometric
line
bundles”
by
the
notation
“V(−)”,
then
we
obtain
a
com-
mutative
diagram
Z
N
⏐
⏐
−→
Y
N
=
Y
N
⏐
⏐
−→
Y
1
⏐
⏐
V(L
N
|
Z
N
)
⏐
⏐
−→
V(L
N
)
⏐
⏐
−→
V(L
⊗N
N
)
⏐
⏐
−→
V(L
1
)
⏐
⏐
Z
N
−→
Y
N
=
Y
N
−→
Y
1
—
where
the
horizontal
morphisms
in
the
first
and
last
lines
are
the
natural
mor-
phisms;
in
the
second
line
of
horizontal
morphisms,
the
first
and
third
horizontal
morphisms
are
the
pull-back
morphisms,
while
the
second
morphism
is
given
by
raising
to
the
N
-th
power;
in
the
first
row
of
vertical
morphisms,
the
morphism
on
the
left
(respectively,
in
the
middle;
on
the
right)
is
that
determined
by
s
N
(respec-
tively,
s
1
|
Y
N
;
s
1
);
the
vertical
morphisms
in
the
second
row
of
vertical
morphisms
are
the
natural
morphisms;
the
vertical
composites
are
the
identity
morphisms.
(ii)
There
is
a
unique
action
of
Π
tp
X
on
L
N
⊗
O
KN
O
J
N
[a
line
bundle
on
Y
N
×
O
KN
O
J
N
]
that
is
compatible
with
the
morphism
Z
N
→
V(L
N
⊗
O
KN
O
J
N
)
determined
by
s
N
[hence
induces
the
identity
on
s
⊗N
=
s
1
|
Z
N
].
Moreover,
this
N
tp
tp
tp
action
of
Π
X
factors
through
Π
X
/Π
Z
N
=
Gal(Z
N
/X),
and,
in
fact,
induces
a
tp
faithful
action
of
Δ
tp
X
/Δ
Z
N
on
L
N
⊗
O
KN
O
J
N
.
Proof.
First,
observe
that
by
the
discussion
above
[concerning
the
structure
of
the
special
fiber
of
Y
N
],
it
follows
that
the
action
of
Π
tp
X
(
Gal(Y
N
/X))
on
Y
N
preserves
the
isomorphism
class
of
the
line
bundle
L
N
,
hence
also
the
isomorphism
class
of
the
line
bundle
L
⊗N
[i.e.,
“the
identification
of
L
⊗N
with
L
1
|
Y
N
,
up
to
N
N
×
×
multiplication
by
an
element
of
Γ(Y
N
,
O
Y
N
)
=
O
K
N
”].
In
particular,
if
we
denote
by
G
N
the
group
of
automorphisms
of
the
pull-back
of
L
N
to
Y
N
×
K
N
J
N
that
lie
over
tp
the
J
N
-linear
automorphisms
of
Y
N
×
K
N
J
N
induced
by
elements
of
Δ
tp
X
/Δ
Y
N
⊆
Gal(Y
N
/X)
and
whose
N
-th
tensor
power
fixes
the
pull-back
of
s
1
|
Y
N
,
then
one
THE
ÉTALE
THETA
FUNCTION
15
verifies
immediately
[by
recalling
the
definition
of
J
N
]
that
G
N
fits
into
a
natural
exact
sequence
tp
1
→
μ
N
(J
N
)
→
G
N
→
Δ
tp
X
/Δ
Y
N
→
1
×
—
where
μ
N
(J
N
)
⊆
J
N
denotes
the
group
of
N
-th
roots
of
unity
in
J
N
.
Now
I
claim
that
the
kernel
H
N
⊆
G
N
of
the
composite
surjection
tp
tp
tp
∼
G
N
Δ
tp
X
/Δ
Y
N
Δ
X
/Δ
Y
=
Z
tp
tp
tp
tp
tp
∼
—
where
we
note
that
Ker(Δ
tp
X
/Δ
Y
N
Δ
X
/Δ
Y
)
=
Δ
Y
/Δ
Y
N
=
Z/N
Z(1)
—
is
an
abelian
group
annihilated
by
multiplication
by
N
.
Indeed,
one
verifies
immediately,
by
considering
various
relevant
line
bundles
on
“G
m
”,
that
[if
we
write
U
for
the
standard
multiplicative
coordinate
on
G
m
and
ζ
for
a
primitive
N
-th
root
of
unity,
then]
this
follows
from
the
identity
of
functions
on
“G
m
”
N−1
f
(ζ
−j
·
U
)
=
1
j=0
def
—
where
f
(U
)
=
(U
−
1)/(U
−
ζ)
represents
an
element
of
H
N
that
maps
to
a
tp
generator
of
Δ
tp
Y
/Δ
Y
N
.
Now
consider
the
tautological
Z/N
Z(1)-torsor
R
N
→
Y
N
obtained
by
extract-
ing
an
N
-th
root
of
s
1
.
More
explicitly,
R
N
→
Y
N
may
be
thought
of
as
the
finite
Y
N
-scheme
associated
to
the
O
Y
N
-algebra
N−1
j=0
L
⊗−j
N
where
the
“algebra
structure”
is
defined
by
the
morphism
L
⊗−N
→
O
Y
N
given
by
N
multiplying
by
s
1
|
Y
N
.
In
particular,
it
follows
immediately
from
the
definition
of
def
G
N
that
G
N
acts
naturally
on
(R
N
)
J
N
=
R
N
×
O
KN
J
N
.
Since
s
1
|
Y
N
has
zeroes
of
order
1
at
each
of
the
cusps
of
Y
N
,
it
thus
follows
immediately
that
(R
N
)
J
N
is
def
connected
and
Galois
over
X
J
N
=
X
×
K
J
N
,
and
that
one
has
an
isomorphism
∼
G
N
→
Gal((R
N
)
J
N
/X
J
N
)
tp
arising
from
the
natural
action
of
G
N
on
(R
N
)
J
N
.
Since
the
abelian
group
Δ
tp
X
/Δ
Y
N
acts
trivially
on
μ
N
(J
N
),
and
H
N
is
annihilated
by
N
,
it
thus
follows
formally
from
the
definition
of
(Δ
X
)
Θ
[i.e.,
as
the
quotient
by
a
certain
“double
commutator
subgroup”]
that
at
least
“geometrically”,
there
exists
a
map
from
Z
N
to
R
N
.
More
precisely,
there
is
a
morphism
Z
N
×
O
JN
K
→
R
N
over
Y
N
.
That
this
morphism
in
fact
factors
through
Z
N
—
inducing
an
isomorphism
∼
Z
N
→
R
N
×
O
KN
O
J
N
16
SHINICHI
MOCHIZUKI
tp
Θ
Θ
—
follows
from
the
definition
of
the
open
immersion
G
J
N
→
(Π
tp
Y
N
)
/N
·
(Δ
Y
)
whose
image
was
used
to
define
Z
N
→
Y
N
[together
with
the
fact
that
s
1
|
Y
N
is
defined
over
K
N
].
This
completes
the
proof
of
assertion
(i).
Next,
we
consider
assertion
(ii).
Since
the
natural
action
of
Π
tp
X
(
Gal(Y
N
/X))
tp
⊗N
on
L
1
|
Y
N
∼
=
L
N
preserves
s
1
|
Y
N
,
and
the
action
of
Π
X
on
Y
N
preserves
the
iso-
morphism
class
of
the
line
bundle
L
N
,
the
existence
and
uniqueness
of
the
desired
action
of
Π
tp
X
on
L
N
⊗
O
KN
O
J
N
follow
immediately
from
the
definitions
[cf.
espe-
cially
the
definition
of
J
N
].
Moreover,
since
s
N
is
defined
over
Z
N
,
it
is
immediate
tp
that
this
action
factors
through
Π
tp
X
/Π
Z
N
.
Finally,
the
asserted
faithfulness
follows
from
the
fact
that
s
1
has
zeroes
of
order
1
at
the
cusps
of
Y
N
[together
with
the
tp
tautological
fact
that
Δ
tp
X
/Δ
Y
N
acts
faithfully
on
Y
N
].
Next,
let
us
set
def
def
J
¨
N
=
K̈
N
(a
1/N
)
a∈
K̈
N
⊆
K
K̈
N
=
K
2N
;
def
Ÿ
N
=
Y
2N
×
O
K̈
O
J
¨
N
;
N
def
Ÿ
N
=
Y
2N
×
K̈
N
J
¨
N
;
L̈
N
=
L
N
|
Ÿ
N
∼
=
L
⊗2
2N
⊗
O
K̈
O
J
¨
N
def
N
and
write
Z̈
N
for
the
composite
of
the
coverings
Ÿ
N
,
Z
N
of
Y
N
;
Z̈
N
for
the
normal-
def
def
def
ization
of
Z
N
in
Z̈
N
;
Ÿ
=
Ÿ
1
=
Y
2
;
Ÿ
=
Ÿ
1
=
Y
2
;
K̈
=
K̈
1
=
J
¨
1
=
K
2
.
Thus,
we
have
a
cartesian
commutative
diagram
V(
L̈
N
)
−→
V(L
N
)
⏐
⏐
⏐
⏐
Ÿ
N
−→
Y
N
—
where,
Π
tp
X
acts
compatibly
on
Ÿ
N
,
Y
N
,
and
[by
Proposition
1.1,
(ii)]
on
L
N
⊗
O
KN
O
J
N
.
Thus,
since
this
diagram
is
cartesian
[and
J
N
⊆
J
¨
N
],
we
ob-
tp
tp
tain
a
natural
action
of
Π
tp
X
on
L̈
N
which
factors
through
Π
X
/Π
Z̈
N
.
Moreover,
we
have
a
natural
exact
sequence
tp
tp
tp
1
→
Π
tp
Z
N
/Π
Z̈
→
Π
Y
/Π
Z̈
→
Gal(Z
N
/Y
)
→
1
N
N
tp
→
Gal(
Ÿ
N
/Y
N
)
—
which
is
compatible
with
the
conjugation
—
where
Π
tp
Z
N
/Π
Z̈
N
actions
by
Π
tp
X
on
each
of
the
terms
in
the
exact
sequence.
Next,
let
us
choose
an
orientation
on
the
dual
graph
of
the
special
fiber
of
Y.
∼
Such
an
orientation
determines
a
specific
isomorphism
Z
→
Z,
hence
a
label
∈
Z
for
each
irreducible
component
of
the
special
fiber
of
Y.
Note
that
this
choice
of
labels
also
determines
a
label
∈
Z
for
each
irreducible
component
of
the
special
fiber
of
Y
N
,
Ÿ
N
.
Now
we
define
D
N
to
be
the
effective
divisor
on
Ÿ
N
which
is
supported
on
the
special
fiber
and
corresponds
to
the
function
Z
j
→
j
2
·
log(q
X
)/2N
THE
ÉTALE
THETA
FUNCTION
17
—
i.e.,
at
the
irreducible
component
labeled
j,
the
divisor
D
N
is
equal
to
the
divisor
j
2
/2N
.
Note
that
since
the
completion
of
Ÿ
N
given
by
the
schematic
zero
locus
of
q
X
at
each
node
of
its
special
fiber
is
isomorphic
to
the
ring
1/2N
O
J
¨
N
[[u,
v]]/(uv
−
q
X
)
—
where
u,
v
are
indeterminates
—
it
follows
that
this
divisor
D
N
is
Cartier.
Moreover,
a
simple
calculation
of
degrees
reveals
that
we
have
an
isomorphism
of
line
bundles
on
Ÿ
N
O
Ÿ
N
(D
N
)
∼
=
L̈
N
—
i.e.,
there
exists
a
section,
well-defined
up
to
an
O
J
×
¨
-multiple,
∈
Γ(
Ÿ
N
,
L̈
N
)
N
whose
zero
locus
is
precisely
the
divisor
D
N
.
That
is
to
say,
we
have
a
commutative
diagram
τ
N
V(
L̈
N
)
−→
V(L
N
)
−→
Ÿ
N
−→
⏐
⏐
⏐
⏐
⏐
⏐
id
Ÿ
N
=
Ÿ
N
−→
Y
N
=
V(L
⊗N
N
)
⏐
⏐
Y
N
−→
V(L
1
)
⏐
⏐
−→
Y
1
in
which
the
second
square
is
the
cartesian
commutative
diagram
discussed
above;
the
third
and
fourth
squares
are
the
lower
second
and
third
squares
of
the
diagram
of
Proposition
1.1,
(i);
τ
N
—
which
we
shall
refer
to
as
the
theta
trivialization
of
L̈
N
—
is
a
section
whose
zero
locus
is
equal
to
D
N
.
Moreover,
since
the
action
of
tp
Π
tp
Y
on
Ÿ
N
clearly
fixes
the
divisor
D
N
,
we
conclude
that
the
action
of
Π
Y
on
Ÿ
N
,
V(
L̈
N
)
always
preserves
τ
N
,
up
to
an
O
J
×
¨
-multiple.
N
Next,
let
M
≥
1
be
an
integer
that
divides
N
.
Then
Y
M
→
Y
(respectively,
Z
M
→
Y;
Ÿ
M
→
Y)
may
be
regarded
as
a
subcovering
of
Y
N
→
Y
(respectively,
⊗N/M
;
Z
N
→
Y;
Ÿ
N
→
Y).
Moreover,
we
have
natural
isomorphisms
L
M
|
Y
N
∼
=
L
N
⊗N/M
.
Thus,
we
obtain
a
diagram
L̈
M
|
Ÿ
N
∼
=
L̈
N
τ
Ÿ
N
⏐
⏐
N
−→
Ÿ
M
M
−→
V(
L̈
M
)
−→
Ÿ
M
V(
L̈
N
)
−→
⏐
⏐
N/M
(−)
Ÿ
N
⏐
⏐
τ
in
which
the
second
square
consists
of
the
natural
morphisms,
hence
commutes;
the
first
square
“commutes
up
to
an
O
J
×
¨
-,
O
J
×
¨
-multiple”,
i.e.,
commutes
up
to
N
M
composition,
at
the
upper
right-hand
corner
of
the
square,
with
an
automorphism
of
V(
L̈
N
)
arising
from
multiplication
by
an
element
of
O
J
×
¨
,
and,
at
the
lower
N
right-hand
corner
of
the
square,
with
an
automorphism
of
V(
L̈
M
)
arising
from
multiplication
by
an
element
of
O
J
×
¨
.
[Indeed,
this
last
commutativity
follows
from
M
the
definition
of
J
¨
N
,
and
the
easily
verified
fact
that
there
exist
“τ
N
’s”
which
are
defined
over
Y
2N
.]
18
SHINICHI
MOCHIZUKI
Thus,
since
by
the
classical
theory
of
the
theta
function
[cf.,
e.g.,
[Mumf],
pp.
306-307;
the
relation
“Θ̈(−
Ü
)
=
−
Θ̈(
Ü)”
given
in
Proposition
1.4,
(ii),
below],
it
follows
that
one
may
choose
τ
1
so
that
the
natural
action
of
Π
tp
Y
on
V(
L̈
1
)
[arising
from
the
fact
that
L̈
1
is
the
pull-back
of
the
line
bundle
L
1
on
Y;
cf.
the
action
of
Proposition
1.1,
(ii)]
preserves
±τ
1
,
we
conclude,
in
light
of
the
definition
of
J
¨
N
,
the
following:
Lemma
1.2.
(Compatibility
of
Theta
Trivializations)
By
modifying
the
⊗N
/N
various
τ
N
by
suitable
O
J
×
¨
-multiples,
one
may
assume
that
τ
N
1
1
2
=
τ
N
2
,
for
all
N
positive
integers
N
1
,
N
2
such
that
N
2
|N
1
.
In
particular,
there
exists
a
compatible
tp
system
[as
N
varies
over
the
positive
integers]
of
actions
of
Π
tp
Y
(respectively,
Π
Ÿ
)
on
Ÿ
N
,
V(
L̈
N
)
which
preserve
τ
N
.
Finally,
each
action
of
this
system
differs
from
the
action
determined
by
the
action
of
Proposition
1.1,
(ii),
by
multiplication
by
a(n)
2N
-th
root
of
unity
(respectively,
N
-th
root
of
unity).
Thus,
by
taking
the
τ
N
to
be
as
in
Lemma
1.2
and
applying
the
natural
iso-
tp
to
the
difference
between
the
actions
of
Π
tp
morphism
Δ
Θ
∼
=
Z(1)
Y
,
Π
Ÿ
arising
from
Proposition
1.1,
(ii),
and
Lemma
1.2,
we
obtain
the
following:
Proposition
1.3.
(The
Étale
Theta
Class)
The
difference
between
the
natural
actions
of
Π
tp
Y
arising
from
Proposition
1.1,
(ii);
Lemma
1.2,
on
constant
multiples
of
τ
N
determines
a
cohomology
class
1
1
Θ
∼
1
tp
η
N
∈
H
1
(Π
tp
Y
,
(
Z/N
Z)(1))
=
H
(Π
Y
,
Δ
Θ
⊗
(
Z/N
Z))
2
2
tp
1
which
arises
from
a
cohomology
class
∈
H
1
(Π
tp
Y
/Π
Z̈
N
,
Δ
Θ
⊗
(
2
Z/N
Z))
whose
re-
striction
to
1
1
tp
tp
tp
Z/N
Z))
=
Hom(Δ
Z/N
Z))
/Δ
,
Δ
⊗
(
/Δ
,
Δ
⊗
(
H
1
(Δ
tp
Θ
Θ
Ÿ
N
Z̈
N
Ÿ
N
Z̈
N
2
2
∼
is
the
composite
of
the
natural
isomorphism
Δ
tp
/Δ
tp
→
Δ
Θ
⊗
Z/N
Z
with
the
Ÿ
Z̈
N
N
natural
inclusion
Δ
Θ
⊗
(Z/N
Z)
→
Δ
Θ
⊗
(
12
Z/N
Z).
Moreover,
if
we
set
×
×
=
{a
∈
O
K̈
|
a
2
∈
K}
O
K/
K̈
def
×
1
as
acting
on
H
1
(Π
tp
and
regard
O
K/
Y
,
(
2
Z/N
Z)(1))
via
the
composite
K̈
1
tp
1
×
1
1
Z/N
Z)(1))
→
H
→
H
(G
,
(
(Π
O
K/
K
Y
,
(
Z/N
Z)(1))
K̈
2
2
—
where
the
first
map
is
the
evident
generalization
of
the
Kummer
map,
which
×
is
compatible
with
the
Kummer
map
O
K
→
H
1
(G
K
,
(Z/N
Z)(1))
relative
to
the
×
×
natural
inclusion
O
K
→
O
K/K̈
and
the
morphism
induced
on
cohomology
by
the
THE
ÉTALE
THETA
FUNCTION
19
natural
inclusion
(Z/N
Z)
→
(
12
Z/N
Z);
the
second
map
is
the
natural
map
—
then
the
set
of
cohomology
classes
1
×
Θ
O
K/
·
η
N
∈
H
1
(Π
tp
Y
,
Δ
Θ
⊗
(
Z/N
Z))
K̈
2
is
independent
of
the
choices
of
s
1
,
s
N
,
τ
N
.
In
particular,
by
allowing
N
to
vary
among
all
positive
integers,
we
obtain
a
set
of
cohomology
classes
tp
1
×
Θ
1
·
η
∈
H
(Π
O
K/
Y
,
Δ
Θ
)
K̈
2
Θ
1
each
of
which
is
a
cohomology
class
∈
H
1
((Π
tp
Y
)
,
2
Δ
Θ
)
whose
restriction
to
tp
Θ
1
Θ
1
H
1
((Δ
tp
Y
)
,
Δ
Θ
)
=
Hom((Δ
Y
)
,
Δ
Θ
)
2
2
∼
Θ
is
the
composite
of
the
natural
isomorphism
(Δ
tp
Y
)
→
Δ
Θ
with
the
natural
inclusion
Δ
Θ
→
12
Δ
Θ
.
Moreover,
the
restricted
classes
1
×
·
η
Θ
|
Ÿ
∈
H
1
(Π
tp
,
Δ
)
O
K/
K̈
Ÿ
2
Θ
arise
naturally
from
classes
×
·
η̈
Θ
∈
H
1
(Π
tp
,
Δ
Θ
)
O
K̈
Ÿ
×
×
—
where
O
K̈
acts
via
the
composite
of
the
Kummer
map
O
K̈
→
H
1
(G
K̈
,
Δ
Θ
)
with
the
natural
map
H
1
(G
K̈
,
Δ
Θ
)
→
H
1
(Π
tp
,
Δ
Θ
)
—
“without
denominators”.
By
Ÿ
×
Θ
abuse
of
the
definite
article,
we
shall
refer
to
any
element
of
the
sets
O
K/
·
η
N
,
K̈
×
×
·
η
Θ
,
O
K̈
·
η̈
Θ
as
the
“étale
theta
class”.
O
K/
K̈
Remark
1.3.1.
Note
that
the
denominators
“
12
”
in
Proposition
1.3
are
by
no
means
superfluous:
Indeed,
this
follows
immediately
from
the
fact
that
the
divisor
D
1
on
Ÿ
clearly
does
not
descend
to
Y.
Let
us
denote
by
U
⊆
Y
the
open
formal
subscheme
obtained
by
removing
the
nodes
from
the
irreducible
component
of
the
special
fiber
labeled
0
∈
Z.
If
we
take
the
unique
cusp
lying
in
U
as
the
origin,
then
—
as
is
well-known
from
the
theory
of
the
Tate
curve
[cf.,
e.g.,
[Mumf],
pp.
306-307]
—
the
group
structure
on
the
underlying
elliptic
curve
of
X
log
determines
a
group
structure
on
U,
together
with
a
unique
[in
light
of
our
choice
of
an
orientation
on
the
dual
graph
of
the
special
fiber
of
Y]
isomorphism
of
U
with
the
p-adic
formal
completion
of
G
m
over
O
K
.
In
particular,
this
isomorphism
determines
a
multiplicative
coordinate
U
∈
Γ(U,
O
U
×
)
20
SHINICHI
MOCHIZUKI
—
which,
as
one
verifies
immediately
from
the
definitions,
admits
a
square
root
Ü
∈
Γ(
Ü,
O
Ü
×
)
def
on
Ü
=
U
×
Y
Ÿ.
Proposition
1.4.
(Relation
to
the
Classical
Theta
Function)
Set
def
−
1
Θ̈
=
Θ̈(
Ü
)
=
q
X
8
·
1
n∈Z
(n+
12
)
2
(−1)
n
·
q
X
2
·
Ü
2n+1
∈
Γ(
Ü,
O
Ü
)
so
Θ̈
extends
uniquely
to
a
meromorphic
function
on
Ÿ
[cf.,
e.g.,
[Mumf],
pp.
306-307].
Then:
(i)
The
zeroes
of
Θ̈
on
Ÿ
are
precisely
the
cusps
of
Ÿ;
each
zero
has
multi-
plicity
1.
The
divisor
of
poles
of
Θ̈
on
Ÿ
is
precisely
the
divisor
D
1
.
(ii)
We
have
Θ̈(
Ü)
=
−
Θ̈(
Ü
−1
);
a/2
Θ̈(−
Ü
)
=
−
Θ̈(
Ü
);
−a
2
/2
Θ̈(q
X
Ü
)
=
(−1)
a
·
q
X
·
Ü
−2a
·
Θ̈(
Ü
)
for
a
∈
Z.
(iii)
The
classes
×
·
η̈
Θ
∈
H
1
(Π
tp
,
Δ
Θ
)
O
K̈
Ÿ
×
-multiples
of
Proposition
1.3
are
precisely
the
“Kummer
classes”
associated
to
O
K̈
of
Θ̈,
regarded
as
a
regular
function
on
Ÿ
.
In
particular,
if
L
is
a
finite
extension
of
K̈,
and
y
∈
Ÿ
(L)
is
a
non-cuspidal
point,
then
the
restricted
classes
×
∼
·
η̈
Θ
|
y
∈
H
1
(G
L
,
Δ
Θ
)
∼
O
K̈
=
H
1
(G
L
,
Z(1))
=
(L
×
)
∧
—
where
the
“∧”
denotes
the
profinite
completion
—
lie
in
L
×
⊆
(L
×
)
∧
and
are
×
×
·
Θ̈(y)
of
Θ̈
and
its
O
K̈
-multiples
at
y.
A
similar
statement
equal
to
the
values
O
K̈
holds
if
y
∈
Ÿ
(L)
is
a
cusp,
if
one
restricts
first
to
the
associated
decomposition
group
D
y
and
then
to
a
section
G
L
→
D
y
compatible
with
the
canonical
integral
structure
[cf.
[Mzk13],
Definition
4.1,
(iii)]
on
D
y
.
In
light
of
this
relationship
between
the
cohomology
classes
of
Proposition
1.3
and
the
values
of
Θ̈,
we
shall
sometimes
refer
to
these
classes
as
“the
étale
theta
function”.
Proof.
Assertion
(ii)
is
a
routine
calculation
involving
the
series
used
to
define
a/2
Θ̈.
A
similar
calculation
shows
that
Θ̈(±1)
=
0.
The
formula
given
for
Θ̈(q
X
Ü
)
in
assertion
(ii)
shows
that
the
portion
of
the
divisor
of
poles
supported
in
the
special
fiber
of
Ÿ
is
equal
to
D
1
.
This
formula
also
shows
that
to
complete
the
proof
of
assertion
(i),
it
suffices
to
show
that
the
given
description
of
the
zeroes
THE
ÉTALE
THETA
FUNCTION
21
and
poles
of
Θ̈
is
accurate
over
the
irreducible
component
of
the
special
fiber
of
Ÿ
labeled
0.
But
this
follows
immediately
from
the
fact
that
the
restriction
of
Θ̈
to
this
irreducible
component
is
the
rational
function
Ü
−
Ü
−1
.
Finally,
in
light
×
×
)
=
O
K
],
assertion
of
assertion
(i)
[and
the
fact,
observed
above,
that
Γ(Y
N
,
O
Y
N
N
×
Θ
(iii)
is
a
formal
consequence
of
the
construction
of
the
classes
O
K̈
·
η̈
.
Proposition
1.5.
(Theta
Cohomology)
(i)
The
Leray-Serre
spectral
sequences
associated
to
the
filtration
of
closed
sub-
groups
tp
Θ
Θ
Δ
Θ
⊆
(Δ
tp
Y
)
⊆
(Π
Y
)
Θ
determine
a
natural
filtration
0
⊆
F
2
⊆
F
1
⊆
F
0
=
H
1
((Π
tp
Y
)
,
Δ
Θ
)
on
the
Θ
cohomology
module
H
1
((Π
tp
Y
)
,
Δ
Θ
)
with
subquotients
·
log(Θ)
F
0
/F
1
=
Hom(Δ
Θ
,
Δ
Θ
)
=
Z
ell
F
1
/F
2
=
Hom((Δ
tp
Y
)
/Δ
Θ
,
Δ
Θ
)
=
Z
·
log(U
)
∼
∼
F
2
=
H
1
(G
K
,
Δ
Θ
)
→
H
1
(G
K
,
Z(1))
→
(K
×
)
∧
—
where
we
use
the
symbol
log(Θ)
to
denote
the
identity
morphism
Δ
Θ
→
Δ
Θ
and
∼
∼
ell
the
symbol
log(U
)
to
denote
the
standard
isomorphism
(Δ
tp
Y
)
/Δ
Θ
→
Z(1)
→
Δ
Θ
.
(ii)
Similarly,
the
Leray-Serre
spectral
sequences
associated
to
the
filtration
of
closed
subgroups
Δ
Θ
⊆
(Δ
tp
)
Θ
⊆
(Π
tp
)
Θ
Ÿ
Ÿ
determine
a
natural
filtration
0
⊆
F̈
2
⊆
F̈
1
⊆
F̈
0
=
H
1
((Π
tp
)
Θ
,
Δ
Θ
)
on
the
Ÿ
)
Θ
,
Δ
Θ
)
with
subquotients
cohomology
module
H
1
((Π
tp
Ÿ
·
log(Θ)
F̈
0
/F̈
1
=
Hom(Δ
Θ
,
Δ
Θ
)
=
Z
·
log(
Ü
)
)
ell
/Δ
Θ
,
Δ
Θ
)
=
Z
F̈
1
/F̈
2
=
Hom((Δ
tp
Ÿ
∼
∼
→
(
K̈
×
)
∧
F̈
2
=
H
1
(G
K̈
,
Δ
Θ
)
→
H
1
(G
K̈
,
Z(1))
def
—
where
we
write
log(
Ü
)
=
12
·
log(U
).
(iii)
Any
class
η̈
Θ
∈
H
1
(Π
tp
,
Δ
Θ
)
arises
from
a
unique
class
[which,
by
abuse
Ÿ
of
notation,
we
shall
denote
by]
η̈
Θ
∈
H
1
((Π
tp
)
Θ
,
Δ
Θ
)
that
maps
to
log(Θ)
in
the
Ÿ
tp
tp
quotient
F̈
0
/F̈
1
and
on
which
a
∈
Z
∼
=
Π
X
/Π
Y
acts
as
follows:
=
Z
∼
η̈
Θ
→
η̈
Θ
−
2a
·
log(
Ü
)
−
a
2
×
·
log(q
X
)
+
log(O
K̈
)
2
—
where
we
use
the
notation
“log”
to
express
the
fact
that
we
wish
to
write
the
group
structure
of
(
K̈
×
)
∧
additively.
Similarly,
any
inversion
automorphism
22
SHINICHI
MOCHIZUKI
ι
of
Π
tp
Y
—
i.e.,
an
automorphism
lying
over
the
action
of
“−1”
on
the
underlying
elliptic
curve
of
X
log
which
fixes
the
irreducible
component
of
the
special
fiber
of
Y
×
×
×
),
but
maps
log(
Ü
)
+
log(O
K̈
)
to
−log(
Ü
)
+
log(O
K̈
).
labeled
0
—
fixes
η̈
Θ
+
log(O
K̈
Proof.
Assertions
(i),
(ii)
follow
immediately
from
the
definitions.
Here,
in
(i)
(respectively,
(ii)),
we
note
that
the
fact
that
F
0
(respectively,
F̈
0
)
surjects
onto
Hom(Δ
Θ
,
Δ
Θ
)
follows,
for
instance,
by
considering
the
Kummer
class
of
the
mero-
morphic
function
Θ̈
·
Ü
−1
on
Y
(respectively,
Θ̈
on
Ÿ
—
cf.
Proposition
1.4,
(iii)).
Assertion
(iii)
follows
from
Propositions
1.3;
1.4,
(ii),
(iii).
Theorem
1.6.
(Tempered
Anabelian
Rigidity
of
the
Étale
Theta
Func-
tion)
Let
X
α
log
(respectively,
X
β
log
)
be
a
smooth
log
curve
of
type
(1,
1)
over
a
finite
extension
K
α
(respectively,
K
β
)
of
Q
p
;
we
assume
that
X
α
log
(respectively,
X
β
log
)
has
stable
reduction
over
O
K
α
(respectively,
O
K
β
),
and
that
the
resulting
special
fibers
are
singular
and
split.
We
shall
use
similar
notation
for
objects
associated
to
X
α
log
,
X
β
log
[but
with
a
subscript
α
or
β]
to
the
notation
that
was
used
for
objects
associated
to
X
log
.
Let
∼
tp
γ
:
Π
tp
X
α
→
Π
X
β
be
an
isomorphism
of
topological
groups.
Then:
)
=
Π
tp
.
(i)
We
have:
γ(Π
tp
Ÿ
Ÿ
α
β
(ii)
γ
induces
an
isomorphism
∼
(Δ
Θ
)
α
→
(Δ
Θ
)
β
that
is
compatible
with
the
surjections
∼
∼
→
(
K̈
α
×
)
∧
Z
H
1
(G
K̈
α
,
(Δ
Θ
)
α
)
→
H
1
(G
K̈
α
,
Z(1))
∼
∼
H
1
(G
K̈
β
,
(Δ
Θ
)
β
)
→
H
1
(G
K̈
β
,
Z(1))
→
(
K̈
β
×
)
∧
Z
determined
by
the
valuations
on
K̈
α
,
K̈
β
.
That
is
to
say,
γ
induces
an
isomor-
∼
phism
H
1
(G
K̈
α
,
(Δ
Θ
)
α
)
→
H
1
(G
K̈
β
,
(Δ
Θ
)
β
)
that
preserves
both
the
kernel
of
these
in
the
resulting
quotients.
surjections
and
the
elements
“1
∈
Z”
(iii)
The
isomorphism
of
cohomology
groups
induced
by
γ
maps
the
classes
×
O
K̈
·
η̈
α
Θ
∈
H
1
(Π
tp
,
(Δ
Θ
)
α
)
Ÿ
α
α
tp
∼
of
Proposition
1.3
for
X
α
to
some
Π
tp
X
β
/Π
Y
β
=
Z-conjugate
of
the
corresponding
classes
×
·
η̈
β
Θ
∈
H
1
(Π
tp
,
(Δ
Θ
)
β
)
O
K̈
Ÿ
β
β
THE
ÉTALE
THETA
FUNCTION
23
of
Proposition
1.3
for
X
β
.
Proof.
Assertion
(i)
is
immediate
from
the
definitions;
the
discreteness
of
the
topo-
tp
logical
group
“Z”;
and
the
fact
that
γ
maps
Δ
tp
X
α
onto
Δ
X
β
[cf.
[Mzk2],
Lemma
1.3.8]
and
preserves
decomposition
groups
of
cusps
[cf.
[Mzk14],
Theorem
6.5,
(iii)].
∼
As
for
assertion
(ii),
the
fact
that
γ
induces
an
isomorphism
(Δ
Θ
)
α
→
(Δ
Θ
)
β
is
immediate
[in
light
of
the
argument
used
to
verify
assertion
(i)]
from
the
defini-
tions.
The
asserted
compatibility
then
follows
from
[Mzk14],
Theorem
6.12;
[Mzk2],
Proposition
1.2.1,
(iv),
(vi),
(vii).
Next,
we
consider
assertion
(iii).
By
composing
γ
with
an
appropriate
in-
ner
automorphism
of
Π
tp
X
β
,
it
follows
from
[Mzk14],
Theorem
6.8,
(ii),
that
we
may
∼
assume
that
the
isomorphism
Π
tp
→
Π
tp
is
compatible
with
suitable
“inversion
au-
Ÿ
Ÿ
α
β
tomorphisms”
ι
α
,
ι
β
[cf.
Proposition
1.5,
(iii)]
on
both
sides.
Next,
let
us
observe
that
it
is
a
tautology
that
γ
is
compatible
with
the
symbols
“log(Θ)”
of
Proposition
1.5,
(i),
(ii).
On
the
other
hand,
by
Proposition
1.5,
(ii),
(iii),
the
property
of
“map-
ping
to
log(Θ)
in
the
quotient
F̈
0
/F̈
1
and
being
fixed,
up
to
a
unit
multiple,
under
an
inversion
automorphism”
completely
determines
the
classes
η̈
Θ
up
to
a
(
K̈
×
)
∧
-
multiple.
Thus,
to
complete
the
proof,
it
suffices
to
reduce
this
“indeterminacy
up
×
to
a
(
K̈
×
)
∧
-multiple”
to
an
“indeterminacy
up
to
a
O
K̈
-multiple”.
×
”
may
be
achieved
[in
This
“reduction
of
indeterminacy
from
(
K̈
×
)
∧
to
O
K̈
light
of
the
compatibility
shown
in
assertion
(ii)]
by
evaluating
the
classes
η̈
Θ
at
a
cusp
that
maps
to
the
irreducible
component
of
the
special
fiber
of
Ÿ
labeled
0
[e.g.,
a
cusp
that
is
preserved
by
the
inversion
automorphism]
via
the
canonical
integral
structure,
as
in
Proposition
1.4,
(iii),
and
applying
the
fact
that,
by
[Mzk14],
Theorem
6.5,
(iii);
[Mzk14],
Corollary
6.11,
γ
preserves
both
the
decomposition
groups
and
the
canonical
integral
structures
[on
the
decomposition
groups]
of
cusps.
Remark
1.6.1.
In
the
proof
of
Theorem
1.6,
(iii),
we
eliminated
the
“indeter-
minacy”
in
question
by
restricting
to
cusps,
via
the
canonical
integral
structure.
Another
way
to
eliminate
this
indeterminacy
is
to
restrict
to
non-cuspidal
torsion
points,
which
are
temp-absolute
by
[Mzk14],
Theorem
6.8,
(iii).
This
latter
ap-
proach
amounts
to
invoking
the
theory
of
[Mzk13],
§2,
which
is,
in
some
sense,
less
elementary
[for
instance,
in
the
sense
that
it
makes
use,
in
a
much
more
essential
way,
of
the
main
result
of
[Mzk11]]
than
the
theory
of
[Mzk13],
§4
[which
one
is,
in
effect,
applying
if
one
uses
cusps].
Remark
1.6.2.
One
way
of
thinking
about
isomorphisms
of
the
tempered
funda-
mental
group
is
that
they
arise
from
variation
of
the
basepoint,
or
underlying
set
theory,
relative
to
which
one
considers
the
associated
“temperoids”
[cf.
[Mzk14],
§3].
Indeed,
this
is
the
point
of
view
taken
in
[Mzk12],
in
the
case
of
anabelioids.
From
this
point
of
view,
the
content
of
Theorem
1.6
may
be
interpreted
as
stating
that:
24
SHINICHI
MOCHIZUKI
The
étale
theta
function
is
preserved
by
arbitrary
“changes
of
the
underly-
ing
set
theory”
relative
to
which
one
considers
the
tempered
fundamental
group
in
question.
When
viewed
in
this
way,
Theorem
1.6
may
be
thought
of
—
especially
if
one
takes
the
non-cuspidal
approach
of
Remark
1.6.1
—
as
a
sort
of
nonarchimedean
analogue
of
the
so-called
functional
equation
of
the
classical
complex
theta
function,
which
also
states
that
the
“theta
function”
is
preserved,
in
effect,
by
“changes
of
the
underlying
set
theory”
relative
to
which
one
considers
the
integral
singular
cohomology
of
the
elliptic
curve
in
question,
i.e.,
more
concretely,
by
the
action
of
the
modular
group
SL
2
(Z).
Note
that
in
the
complex
case,
it
is
crucial,
in
order
to
prove
the
functional
equation,
to
have
not
only
the
“Schottky
uniformization”
C
×
C
×
/q
Z
by
C
×
—
which
naturally
gives
rise
to
the
analytic
series
representation
of
the
theta
function,
but
is
not,
however,
preserved
by
the
action
of
the
modular
group
—
but
also
the
full
uniformization
of
an
elliptic
curve
by
C
[which
is
preserved
by
the
action
of
the
modular
group].
This
“preservation
of
the
full
uniformization
by
C”
in
the
complex
case
may
be
regarded
as
being
analogous
to
the
preservation
of
the
non-cuspidal
torsion
points
in
the
approach
to
proving
Theorem
1.6
discussed
in
Remark
1.6.1.
Remark
1.6.3.
The
interpretation
of
Theorem
1.6
given
in
Remark
1.6.2
is
rem-
iniscent
of
the
discussion
given
in
the
Introduction
of
[Mzk7],
in
which
the
author
expresses
his
hope,
in
effect,
that
some
sort
of
p-adic
analogue
of
the
functional
equation
of
the
theta
function
could
be
developed.
Remark
1.6.4.
One
verifies
immediately
that
there
are
[easier]
profinite
versions
of
the
constructions
given
in
the
present
§1:
That
is
to
say,
if
we
denote
by
(
Ÿ
log
)
∧
→
(Y
log
)
∧
→
X
log
the
profinite
étale
coverings
determined
by
the
tempered
coverings
Ÿ
log
→
Y
log
→
then
the
set
of
classes
O
×
·
η̈
Θ
∈
H
1
(Π
tp
,
Δ
)
determines,
X
log
[so
Π
X
/Π
Y
∧
∼
=
Z],
Θ
Ÿ
K̈
by
profinite
completion,
a
set
of
classes
×
O
K̈
·
(η̈
Θ
)
∧
∈
H
1
(Π
Ÿ
∧
,
Δ
Θ
)
on
which
any
Π
X
/Π
Y
∧
∼
=
Z
a
acts
via
a
2
×
(η̈
)
→
(η̈
)
−
2a
·
log(
Ü
)
−
)
·
log(q
X
)
+
log(O
K̈
2
Θ
∧
Θ
∧
[cf.
Proposition
1.5,
(iii)].
Moreover,
given
X
α
,
X
β
as
in
Theorem
1.6,
any
isomor-
phism
∼
Π
X
α
→
Π
X
β
preserves
these
profinite
étale
theta
functions.
THE
ÉTALE
THETA
FUNCTION
25
×
In
fact,
it
is
possible
to
eliminate
the
O
K̈
-indeterminacy
of
Theorem
1.6,
(iii),
to
a
substantial
extent
[cf.
[Mzk13],
Corollary
4.12].
For
simplicity,
let
us
assume
in
the
following
discussion
that
the
following
two
conditions
hold:
(I)
K
=
K̈.
(II)
The
hyperbolic
curve
determined
by
X
log
is
not
arithmetic
over
K
[cf.,
e.g.,
[Mzk3],
Remark
2.1.1].
As
is
well-known,
condition
(II)
amounts,
relative
to
the
j-invariant
of
the
elliptic
curve
underlying
X,
to
the
assertion
that
we
exclude
four
exceptional
j-invariants
[cf.
[Mzk3],
Proposition
2.7].
Now
let
us
write
Ẍ
log
→
X
log
for
the
Galois
[by
condition
(I)]
covering
of
degree
4
determined
by
the
“multiplication
by
2”
map
on
the
elliptic
curve
underlying
X;
write
X
log
→
C
log
for
the
stack-theoretic
quotient
of
X
log
by
the
natural
action
of
±1
on
[the
underlying
elliptic
curve
of]
X.
Thus,
[by
condition
(II)]
the
hyperbolic
orbicurve
determined
by
C
log
is
a
K-core
[cf.
[Mzk3],
Remark
2.1.1].
Observe
that
the
covering
Ẍ
log
→
C
log
is
Galois,
with
Galois
group
isomorphic
to
(Z/2Z)
3
.
Moreover,
we
have
two
natural
automorphisms
μ
∈
Gal(
Ẍ/X)
⊆
Gal(
Ẍ/C);
±
∈
Gal(
Ẍ/C)
—
i.e.,
respectively,
the
unique
nontrivial
element
of
Gal(
Ẍ/X)
that
acts
trivially
on
the
set
of
irreducible
components
of
the
special
fiber;
the
unique
nontrivial
element
of
Gal(
Ẍ/C)
that
acts
trivially
on
the
set
of
cusps
of
Ẍ.
Now
suppose
that
we
are
given
a
nontrivial
element
Z
∈
Gal(
Ẍ/X)
which
is
=
μ
.
Then
Z
determines
a
commutative
diagram
Ÿ
log
−→
Ẍ
log
−→
Ẋ
log
⏐
⏐
Ċ
log
−→
X
log
⏐
⏐
−→
C
log
—
where
Ÿ
log
→
Ẍ
log
,
X
log
→
C
log
are
the
natural
morphisms;
Ẍ
log
→
Ẋ
log
is
the
quotient
by
the
action
of
Z
;
Ẋ
log
→
X
log
is
the
quotient
by
the
action
of
μ
;
Ẋ
log
→
Ċ
log
is
the
[stack-theoretic]
quotient
by
the
action
of
±
·
μ
;
the
square
is
cartesian.
Definition
1.7.
We
shall
refer
to
a
smooth
log
orbicurve
over
K
that
arises,
up
to
isomorphism,
as
the
smooth
log
orbicurve
Ẋ
log
(respectively,
Ċ
log
)
constructed
above
for
some
choice
of
Z
as
being
of
type
(1,
μ
2
)
(respectively,
(1,
μ
2
)
±
).
We
shall
also
apply
this
terminology
to
the
associated
hyperbolic
orbicurves.
26
SHINICHI
MOCHIZUKI
Proposition
1.8.
(Characteristic
Nature
of
Coverings)
For
=
α,
β,
let
log
Ẋ
be
a
smooth
log
curve
of
type
(1,
μ
2
)
over
a
finite
extension
K
of
Q
p
;
log
log
log
log
,
X
,
Ċ
,
C
for
the
related
smooth
log
orbicurves
[as
in
the
write
Ÿ
log
,
Ẍ
above
discussion].
Then
any
isomorphism
of
topological
groups
∼
∼
→
Π
tp
(respectively,
γ
:
Π
tp
→
Π
tp
)
γ
:
Π
tp
Ẋ
Ẋ
Ċ
Ċ
α
α
β
β
induces
an
isomorphism
between
the
commutative
diagrams
of
outer
homomor-
phisms
of
topological
groups
Π
tp
Ÿ
−→
Π
tp
Ẍ
−→
Π
tp
Ẋ
⏐
⏐
−→
Π
tp
X
⏐
⏐
Π
tp
Ċ
−→
Π
tp
C
—
where
=
α,
β.
A
similar
statement
holds
when
“Π
tp
”
is
replaced
by
“Π”.
Proof.
First,
we
consider
the
tempered
case.
By
[Mzk2],
Lemma
1.3.8,
it
follows
that
γ
is
compatible
with
the
respective
projections
to
G
K
.
By
[Mzk14],
Theorem
6.8,
(ii)
[cf.
also
[Mzk3],
Theorem
2.4],
it
follows
from
condition
(II)
that
γ
induces
∼
tp
an
isomorphism
Π
tp
C
α
→
Π
C
β
that
is
compatible
with
γ.
Since
[as
is
easily
verified]
tp
Δ
tp
X
⊆
Δ
C
may
be
characterized
as
the
unique
open
subgroup
of
index
2
that
corresponds
to
a
double
covering
which
is
a
scheme
[i.e.,
open
subgroup
of
index
2
whose
profinite
completion
contains
no
torsion
elements
—
cf.,
e.g.,
[Mzk16],
∼
tp
Lemma
2.1,
(v)],
it
follows
that
γ
determines
an
isomorphism
Δ
tp
X
α
→
Δ
X
β
,
hence
∼
ell
ell
also
an
isomorphism
(Δ
tp
→
(Δ
tp
X
α
)
X
β
)
.
Moreover,
by
considering
the
discrete-
ness
of
Gal(Y
/X
)
∼
=
Z
,
or,
alternatively,
the
triviality
of
the
action
of
G
K
on
Gal(Y
/X
),
it
follows
that
this
last
isomorphism
determines
an
isomorphism
tp
∼
tp
tp
∼
Δ
tp
X
α
/Δ
Y
α
→
Δ
X
β
/Δ
Y
β
=
Z
β
,
hence
[by
considering
the
kernel
of
the
action
of
∼
tp
tp
tp
tp
log
log
Π
tp
C
on
Δ
X
/Δ
Y
]
an
isomorphism
Π
X
α
→
Π
X
β
.
Since
Ẋ
→
Ċ
may
be
char-
log
log
acterized
as
the
quotient
by
the
unique
automorphism
of
Ẋ
over
C
that
acts
log
nontrivially
on
the
cusps
of
Ẋ
[where
we
recall
that
γ
preserves
decomposition
log
,
we
group
of
cusps
—
cf.
[Mzk14],
Theorem
6.5,
(iii)]
but
does
not
lie
over
X
tp
tp
thus
conclude
that
γ
induces
isomorphisms
between
the
respective
Π
Ẋ
,
Π
Ċ
,
Π
tp
X
,
Π
tp
C
that
are
compatible
with
the
natural
inclusions
among
these
subgroups
[for
a
fixed
“”].
Moreover,
since
γ
preserves
the
decomposition
groups
of
cusps
of
Π
tp
X
[cf.
[Mzk14],
Theorem
6.5,
(iii)],
we
conclude
immediately
that
γ
is
also
compatible
⊆
Π
tp
⊆
Π
tp
with
the
subgroups
Π
tp
X
,
as
desired.
The
profinite
case
is
proven
Ÿ
Ẍ
similarly
[or
may
be
derived
from
the
tempered
case
via
[Mzk14],
Theorem
6.6].
Next,
let
us
suppose
that
√
−1
∈
K
THE
ÉTALE
THETA
FUNCTION
27
√
—
where
we
note
that
“
−1”
determines
a
4-torsion
point
τ
of
[the
underlying
elliptic
curve
of]
Ẋ
whose
restriction
to
the
special
fiber
lies
in
the
interior
of
[i.e.,
avoids
the
nodes
of]
the
unique
irreducible
component
of
the
special
fiber;
the
√
−1
4-torsion
point
“τ
”
determined
by
“−
−1”
admits
a
similar
description.
Let
,
Δ
Θ
)
η̈
Θ
∈
H
1
(Π
tp
Ÿ
∼
/Π
tp
be
a
class
as
in
Proposition
1.3;
write
η̈
Θ,Z
for
the
Π
tp
=
Z-orbit
of
η̈
Θ
.
Ẋ
Ÿ
Definition
1.9.
Suppose
that
√
−1
∈
K.
(i)
We
shall
refer
to
either
of
the
following
two
sets
of
values
[cf.
Proposition
1.4,
(iii)]
of
η̈
Θ,Z
η̈
Θ,Z
|
τ
,
η̈
Θ,Z
|
τ
−1
⊆
K
×
as
a
standard
set
of
values
of
η̈
Θ,Z
.
×
(ii)
If
η̈
Θ,Z
satisfies
the
property
that
the
unique
value
∈
O
K
[cf.
the
value
√
at
−1
of
the
series
representation
of
Θ̈
given
in
Proposition
1.4;
Proposition
1.4,
(ii)]
of
maximal
order
[i.e.,
relative
to
the
valuation
on
K]
of
some
standard
set
of
values
of
η̈
Θ,Z
is
equal
to
±1,
then
we
shall
say
that
η̈
Θ,Z
is
of
standard
type.
Remark
1.9.1.
Observe
that
it
is
immediate
from
the
definitions
that
any
inner
automorphism
of
Π
tp
arising
from
Π
tp
acts
trivially
on
η̈
Θ,Z
,
and
that
the
Ċ
Ẋ
automorphisms
μ
,
±
map
η̈
Θ,Z
→
−η̈
Θ,Z
[cf.
Proposition
1.4,
(ii)].
In
particular,
maps
η̈
Θ,Z
→
η̈
Θ,Z
.
any
inner
automorphism
of
Π
tp
Ċ
The
point
of
view
of
Remark
1.6.1
motivates
the
following
result:
Theorem
1.10.
(Constant
Multiple
Rigidity
of
the
Étale
Theta
Func-
log
tion)
For
=
α,
β,
let
Ċ
be
a
smooth
log
curve
of
type
(1,
μ
2
)
±
over
a
finite
extension
K
of
Q
p
that
contains
a
square
root
of
−1.
Let
∼
γ
:
Π
tp
→
Π
tp
Ċ
Ċ
α
β
∼
tp
be
an
isomorphism
of
topological
groups.
Suppose
that
the
isomorphism
Π
tp
X
α
→
Π
X
β
Θ,Z
induced
by
γ
[cf.
Proposition
1.8]
maps
η̈
α
Θ,Z
→
η̈
β
[cf.
Theorem
1.6,
(iii)].
Then:
Θ,Z
(i)
The
isomorphism
γ
preserves
the
property
that
η̈
be
of
standard
type,
a
property
that
determines
this
collection
of
classes
up
to
multiplication
by
±1.
(ii)
The
isomorphism
∼
K
α
×
→
K
β
×
28
SHINICHI
MOCHIZUKI
×
×
∧
×
∧
∼
—
where
we
regard
K
⊆
(K
)
as
a
subset
of
(K
)
=
H
1
(G
K
,
(Δ
Θ
)
)
⊆
,
(Δ
Θ
)
)
—
induced
by
[an
arbitrary]
γ
preserves
the
standard
sets
of
H
1
(Π
tp
Ċ
Θ,Z
values
of
η̈
.
Θ,Z
(iii)
Suppose
that
η̈
is
of
standard
type,
and
that
the
residue
charac-
Θ,Z
teristic
of
K
is
odd.
Then
η̈
determines
a
{±1}-structure
[cf.
[Mzk13],
×
∧
Corollary
4.12;
Remark
1.10.1,
(ii),
below]
on
the
(K
)
-torsor
at
the
unique
cusp
log
of
Ċ
that
is
compatible
with
the
canonical
integral
structure
and,
moreover,
preserved
by
[arbitrary]
γ.
Proof.
First,
we
observe
that
assertions
(i),
(iii)
follow
formally
from
assertion
(ii)
[cf.
also
the
series
representation
of
Proposition
1.4].
Now
we
verify
assertion
(ii),
as
follows.
By
applying
[Mzk14],
Theorem
6.8,
(iii),
together
with
the
fact
that
γ
induces
isomorphisms
between
the
dual
graphs
[cf.
Theorem
1.6,
(i);
[Mzk2],
Lemma
2.3]
of
the
special
fibers
of
the
various
corresponding
coverings
of
C
α
,
C
β
that
appear
in
the
proof
of
[Mzk14],
Theorem
6.8,
(iii),
it
follows
that
γ
maps
[the
decomposition
group
of
points
of
Ÿ
α
lying
over]
τ
to
[the
decomposition
group
of
points
of
Ÿ
β
lying
over]
τ
±1
.
Now
assertion
(ii)
follows
immediately.
Remark
1.10.1.
(i)
The
“±-indeterminacy”
of
Theorem
1.10,
(i),
(iii),
is
reminiscent
of,
but
stronger
than,
the
indeterminacy
up
to
multiplication
by
a
12-th
root
of
unity
of
[Mzk13],
Corollary
4.12.
Also,
we
note
that
from
the
point
of
view
of
the
technique
of
the
proof
of
loc.
cit.,
applied
in
the
present
context
of
“Tate
curves”,
it
is
the
fact
that
there
is
a
“special
2-torsion
point”,
i.e.,
the
2-torsion
point
whose
image
in
the
special
fiber
lies
in
the
same
irreducible
component
as
the
origin,
that
allows
one
to
reduce
the
“12
=
2
·
(3!)”
of
loc.
cit.
to
“2”.
(ii)
We
take
this
opportunity
to
remark
that
in
[Mzk13],
Corollary
4.12,
the
author
omitted
the
hypothesis
that
“K
contain
a
primitive
12-th
root
of
unity”.
The
author
apologizes
for
this
omission.
Remark
1.10.2.
One
observation
that
one
might
make
is
that
since
Theorem
1.10
depends
on
the
theory
of
[Mzk13],
§2
[i.e.,
in
the
“tempered
version”
of
this
theory
given
in
[Mzk14],
§6],
one
natural
approach
to
further
strengthening
Theo-
rem
1.10
is
to
consider
applying
the
“absolute
p-adic
version
of
the
Grothendieck
Conjecture”
of
[Mzk16],
Corollary
2.3,
which
may
be
regarded
as
a
“strengthening”
of
the
theory
of
[Mzk13],
§2.
One
problem
here,
however,
is
that
unlike
the
portion
of
the
theory
of
[Mzk13],
§2,
that
concerns
[non-cuspidal]
torsion
points
of
once-
punctured
elliptic
curves
[i.e.,
[Mzk13],
Corollary
2.6],
the
“absolute
p-adic
version
of
the
Grothendieck
Conjecture”
of
[Mzk16],
Corollary
2.3,
only
holds
for
elliptic
curves
which
are
defined
over
number
fields.
Moreover,
even
if,
in
the
future,
this
hypothesis
should
be
eliminated,
the
[somewhat
weaker]
theory
of
[Mzk13],
§2,
fol-
lows
“formally”
[cf.
[Mzk13],
Remark
2.8.1]
from
certain
“general
nonsense”-type
THE
ÉTALE
THETA
FUNCTION
29
arguments
that
hold
over
any
base
over
which
the
relative
isomorphism
version
of
the
Grothendieck
Conjecture
[i.e.,
the
isomorphism
portion
of
[Mzk11],
Theorem
A],
together
with
the
absolute
preservation
of
cuspidal
decomposition
groups
[cf.
[Mzk13],
Theorem
1.3,
(iii)],
holds.
In
particular,
by
restricting
our
attention
to
consequences
of
this
“general
nonsense”
in
the
style
of
[Mzk13],
§2,
one
may
hope
to
generalize
the
results
discussed
in
the
present
§1
to
much
more
general
bases
[such
as,
for
instance,
Z
p
[[q]][q
−1
]
⊗
Q
p
,
where
q
is
an
indeterminate
intended
to
suggest
the
“q-parameter
of
a
Tate
curve”],
or,
for
instance,
to
the
case
of
“pro-Σ
versions
of
the
tempered
fundamental
group”
[i.e.,
where
Σ
is
a
set
of
primes
containing
p
which
is
not
necessarily
the
set
of
all
prime
numbers]
—
situations
in
which
it
is
by
no
means
clear
[at
least
at
the
time
of
writing]
whether
or
not
it
is
possible
to
prove
an
“absolute
version
of
the
Grothendieck
Conjecture”.
Remark
1.10.3.
(i)
Note
that
the
étale
theta
function
arises
as
a
cohomology
class
of
a
certain
subgroup
of
the
“theta
quotient”
tp
Θ
Π
tp
X
(Π
X
)
[cf.
Propositions
1.3,
1.5].
On
the
other
hand,
the
very
strong
rigidity
property
of
Theorem
1.10,
(i),
clearly
—
as
one
may
see,
for
instance,
by
considering
au-
Θ
tomorphisms
of
the
topological
group
(Π
tp
—
fails
to
hold
if,
for
instance,
one
X
)
tp
Θ
replaces
“Π
C
”
by
the
corresponding
“theta
quotient”
(Π
tp
C
)
.
Thus,
even
though
at
first
glance,
the
theory
of
the
étale
theta
function
may
appear
only
to
involve
tp
Θ
the
theta
quotient
(Π
tp
X
)
,
in
fact,
the
full
tempered
fundamental
group
Π
X
plays
an
essential
role
in
the
theory
of
rigidity
properties
of
the
étale
theta
function.
(ii)
Relative
to
the
eventual
goal
of
applying
the
theory
of
the
present
paper
to
Hodge-Arakelov
theory
[cf.
the
Introduction]
—
which
concerns
restricting
the
theta
function
and
its
derivatives
to
torsion
points
—
the
necessity
of
working
with
the
full
tempered
fundamental
group
may
be
motivated
geometrically
as
follows.
The
hyperbolic
orbicurves
involved
may
be
thought
of,
up
to
isogeny
[cf.
§0],
as
the
hyperbolic
curves
obtained
by
removing
the
l-torsion
points
[for
some
integer
l
≥
1]
from
an
elliptic
curve
[cf.
especially
the
theory
of
§2
below].
At
the
level
of
topological
surfaces,
the
complement
S
ell\tors
of
the
l-torsion
points
in
an
elliptic
curve
E
may
be
thought
of
as
the
result
S
ell\tors
=
S
ell\disc
S
disc\tors
of
gluing
the
complement
S
ell\disc
of
a
single
disc
in
an
elliptic
curve
to
the
comple-
ment
S
disc\tors
in
a
disc
of
a
finite
collection
of
points
[i.e.,
where
the
gluing
is
along
the
circle
which
forms
the
boundary
of
the
discs
involved].
The
topological
funda-
mental
group
of
S
ell\tors
may
then
be
thought
of
as
the
amalgamated
sum
of
the
fundamental
groups
of
S
ell\disc
,
S
disc\tors
.
Now
the
fundamental
group
of
S
ell\disc
is
a
free
group
on
two
generators,
which
may
be
thought
of
as
a
basis
of
the
abelian
fundamental
group
of
the
original
elliptic
curve
E.
This
portion
of
the
fundamental
30
SHINICHI
MOCHIZUKI
group
of
S
ell\tors
may
be
thought
of
as
corresponding
to
the
“Heisenberg
group”,
or
theta-group
—
i.e.,
the
theta
quotient
discussed
in
(i)
—
which
plays
a
fundamental
role
in
the
theory
of
theta
functions,
hence
in
the
scheme-theoretic
Hodge-Arakelov
theory
reviewed
in
[Mzk4]
[cf.
especially
the
discussion
of
[Mzk4],
§1.3.5].
On
the
other
hand,
S
disc\tors
is
homotopically
equivalent
to
a
bouquet
of
circles,
where
the
circles
correspond
naturally
to
the
l-torsion
points
of
E.
Thus,
from
the
point
of
view
of
algebraic
topology,
the
S
disc\tors
portion
of
S
ell\tors
may
be
thought
of
as
the
suspension
of
the
discrete
set
of
l-torsion
points
[together
with
some
additional
“basepoint”].
On
the
other
hand,
from
the
point
of
view
of
arithmetic
geometry,
the
circle
may
be
thought
of
as
a
sort
of
“Tate
motive”
and
the
“arithmetic
suspen-
sion”
constituted
by
the
S
disc\tors
portion
of
S
ell\tors
as
a
sort
of
“anabelian
Tate
twist”
of
the
l-torsion
points
of
E.
Moreover,
from
the
point
of
view
of
Hodge-
Arakelov
theory,
the
universal
covering
of
the
bouquet
constituted
by
S
disc\tors
may
be
thought
of
as
the
result
of
“filling
in
the
discrete
set
of
l-torsion
points”
by
joining
these
points
via
continuous
line
segments
—
cf.
the
original
point
of
view
of
Hodge-Arakelov
theory
[discussed
in
[Mzk4],
§1.3.4]
to
the
effect
that
the
set
of
torsion
points
is
to
be
regarded
as
a
“high
resolution
approximation”
of
the
under-
lying
real
analytic
manifold
of
E.
Put
another
way,
this
sort
of
“continuous
version
of
the
l-torsion
points”
may
be
thought
of
as
a
sort
of
“cyclotomic
blurring”
of
the
l-torsion
points
—
i.e.,
a
sort
of
space
of
infinitesimal
deformations
of
the
l-torsion
points
that
allows
one
to
consider
derivatives
[e.g.,
of
the
theta
function]
at
the
l-torsion
points.
Thus,
in
summary,
relative
to
these
geometric
considerations,
the
S
disc\tors
,
hence
also
its
group-theoretic
topological
surface
S
ell\tors
=
S
ell\disc
counterpart
[i.e.,
the
full
fundamental
group
under
consideration]
may
be
thought
of
as
being
precisely
a
geometric
realization
of
the
content
(theta
functions
and
their
derivatives)|
l-torsion
points
[i.e.,
where
“theta
functions”
correspond,
via
theta-groups,
to
S
ell\disc
,
and
“deriva-
tives
at
l-torsion
points”
correspond
to
S
disc\tors
]
of
Hodge-Arakelov
theory.
Remark
1.10.4.
(i)
As
observed
in
Proposition
1.4,
(iii),
by
restricting
the
étale
theta
function
to
various
[e.g.,
torsion!]
points,
one
obtains
a
Kummer-theoretic
approach
to
considering
the
theta
function
as
a
function
on
points.
In
the
context
of
the
theory
of
the
present
§1
[and
indeed
of
the
present
paper],
in
which
one
does
not
assume
[cf.
Remark
1.10.2]
that
the
multiplicative
groups
associated
to
the
base
fields
involved
that
appear
in
the
absolute
Galois
groups
of
these
fields
—
e.g.,
the
“(L
×
)
∧
”
that
appears
in
Proposition
1.4,
(iii)
—
are
equipped
with
additive
structures
[i.e.,
arising
from
the
addition
operation
on
the
field],
the
functions
that
may
be
obtained
in
this
way
are
very
special.
Indeed,
if
one
may
avail
oneself
of
both
the
additive
and
multiplicative
structures
—
i.e.,
the
ring
structures
—
of
the
fields
involved,
then
it
is
not
difficult
to
give
various
“group-theoretic
algorithms”
for
constructing
all
sorts
of
such
“functions”.
On
the
other
hand:
If
one
may
only
avail
oneself
of
the
multiplicative
structure,
then
it
is
difficult
to
construct
such
functions,
except
via
considering
directly
the
THE
ÉTALE
THETA
FUNCTION
31
functions
obtained
by
restricting
Kummer
classes
of
meromorphic
func-
tions.
[Of
course,
the
multiplicative
structure
allows
one
to
construct
N
-th
powers,
for
N
≥
1
an
integer,
of
the
values
obtained
by
restricting
Kummer
classes,
but
such
N
-th
powers
are
simply
the
values
obtained
by
restricting
the
Kummer
classes
obtained
by
multiplying
the
original
Kummer
classes
by
N
.]
Finally,
we
recall
that
the
theta
function
satisfies
the
unusual
property,
among
meromorphic
functions
on
tempered
coverings
of
pointed
stable
curves,
of
having
a
divisor
of
poles
that
is
contained
in
the
special
fiber
and
a
divisor
zeroes
that
does
not
contain
any
irreducible
components
of
the
special
fiber
[cf.
Proposition
1.4,
(i)].
(ii)
Even
if
one
allows
oneself
to
consider
“Kummer
classes”
κ
∈
H
1
(Π,
Z(1))
for
Π
an
arbitrary
topological
group
that
surjects
onto
the
absolute
Galois
group
G
K
of
a
finite
extension
K
of
Q
p
,
it
is
not
difficult
to
see
that
it
is
a
highly
nontrivial
operation
to
construct
functions
on
the
set
of
[conjugacy
classes
of]
sections
of
Π
G
K
.
Indeed,
if
κ
is
to
give
rise
to
a
nontrivial
function,
then
it
is
natural
to
assume
that
it
must
induce
an
isomorphism
of
some
isomorph
[cf.
§0]
of
Z(1)
inside
Π
onto
Z(1)
[cf.
the
role
of
“Δ
Θ
”
in
Proposition
1.3
in
the
case
of
the
étale
theta
function].
On
the
other
hand,
if,
for
instance,
Π
=
Z(1)
G
K
,
then
[cf.
the
discussion
of
the
“theta
quotient”
in
Remark
1.10.3,
(i)]
it
is
easy
to
see
that
the
resulting
κ
fails
to
satisfy
the
analogue
of
the
rigidity
property
of
Theorem
1.10,
(i).
Thus,
just
as
in
the
discussion
of
Remark
1.10.3,
one
is
ultimately
led
naturally
to
consider
the
case
where
Π
is
some
sort
of
[e.g.,
tempered
or
profinite]
arithmetic
group
of
a
hyperbolic
orbicurve,
in
which
case
various
strong
anabelian
rigidity
properties
are
known.
Moreover,
it
is
difficult
to
see
how
to
develop
the
theory
of
§2
below
—
which
makes
essential
use,
in
so
many
ways,
of
the
theory
and
structure
of
theta-groups
[i.e.,
the
“theta
quotients”
of
Remark
1.10.3,
(i)]
—
for
hyperbolic
orbicurves
that
are
not
isogenous
to
once-punctured
elliptic
curves.
This
state
of
affairs
again
serves
to
highlight
the
fact
that
the
function
[on,
say,
torsion
points]
determined
by
the
étale
theta
function
should
be
regarded
as
a
very
special
and
unusual
object.
32
SHINICHI
MOCHIZUKI
Section
2:
The
Theory
of
Theta
Environments
In
this
§,
we
begin
by
discussing
various
“general
nonsense”
complements
[cf.
Corollaries
2.8,
2.9]
to
the
theory
of
étale
theta
functions
of
§1
involving
coverings
[cf.
the
discussion
of
the
“Lagrangian
approach
to
Hodge-Arakelov
theory”
in
the
Introduction].
This
discussion
leads
naturally
to
the
theory
of
the
cyclotomic
envelope
[cf.
Definition
2.10]
and
the
associated
mono-
and
bi-theta
environments
[cf.
Definition
2.13],
whose
tempered
anabelian
rigidity
properties
[cf.
Corollaries
2.18,
2.19]
we
shall
use
in
§5,
below,
to
relate
the
theory
of
the
present
§2
to
the
theory
of
tempered
Frobenioids
to
be
discussed
in
§3,
§4,
below.
Let
X
log
be
a
smooth
log
curve
of
type
(1,
1)
over
a
field
K
of
characteristic
zero.
For
simplicity,
we
assume
that
the
hyperbolic
curve
determined
by
X
log
is
not
K-arithmetic
[i.e.,
admits
a
K-core
—
cf.
[Mzk3],
Remark
2.1.1].
As
in
§1,
we
shall
denote
the
(profinite)
étale
fundamental
group
of
X
log
by
Π
X
.
Thus,
we
have
a
natural
exact
sequence:
1
→
Δ
X
→
Π
X
→
G
K
→
1
def
—
where
G
K
=
Gal(K/K);
Δ
X
is
defined
so
as
to
make
the
sequence
exact.
Since
Δ
X
is
a
profinite
free
group
on
2
generators,
the
quotient
def
Δ
Θ
X
=
Δ
X
/[Δ
X
,
[Δ
X
,
Δ
X
]]
fits
into
a
natural
exact
sequence
ell
1
→
Δ
Θ
→
Δ
Θ
X
→
Δ
X
→
1
def
ab
ell
2
—
where
Δ
ell
X
=
Δ
X
=
Δ
X
/[Δ
X
,
Δ
X
];
we
write
Δ
Θ
for
the
image
of
∧
Δ
X
in
Θ
Δ
Θ
X
.
Also,
we
shall
write
Π
X
Π
X
for
the
quotient
whose
kernel
is
the
kernel
of
the
quotient
Δ
X
Δ
Θ
X
.
Now
let
l
≥
1
be
an
integer.
One
verifies
easily
by
considering
the
well-known
Θ
structure
of
Δ
Θ
X
that
the
subgroup
of
Δ
X
generated
by
l-th
powers
of
elements
of
Θ
Θ
Δ
Θ
X
is
normal.
We
shall
write
Δ
X
Δ
X
for
the
quotient
of
Δ
X
by
this
normal
subgroup.
Thus,
the
above
exact
sequence
for
Δ
Θ
X
determines
a
quotient
exact
sequence
ell
1
→
Δ
Θ
→
Δ
X
→
Δ
X
→
1
ell
—
where
Δ
Θ
∼
=
(Z/lZ)(1);
Δ
X
is
a
free
(Z/lZ)-module
of
rank
2.
Also,
we
shall
write
Π
X
Π
X
for
the
quotient
whose
kernel
is
the
kernel
of
the
quotient
Δ
X
ell
def
Δ
X
and
Π
X
=
Π
X
/Δ
Θ
.
Let
us
write
x
for
the
unique
cusp
of
X
log
.
Then
there
is
a
natural
injective
[outer]
homomorphism
D
x
→
Π
Θ
X
THE
ÉTALE
THETA
FUNCTION
33
—
where
D
x
⊆
Π
X
is
the
decomposition
group
associated
to
x
—
which
maps
the
inertia
group
I
x
⊆
D
x
isomorphically
onto
Δ
Θ
.
Thus,
we
have
exact
sequences
1
→
Δ
X
→
Π
X
→
G
K
→
1;
1
→
Δ
Θ
→
D
x
→
G
K
→
1
—
where
we
write
D
x
⊆
Π
X
for
the
image
of
D
x
in
Π
X
.
ell
Next,
let
Π
X
Q
be
a
quotient
onto
a
free
(Z/lZ)-module
Q
of
rank
1
such
ell
that
the
restricted
map
Δ
X
→
Q
is
still
surjective,
but
the
restricted
map
D
x
→
Q
is
trivial.
Denote
the
corresponding
covering
by
X
log
→
X
log
;
write
Π
X
⊆
Π
X
,
ell
ell
Δ
X
⊆
Δ
X
,
Δ
X
⊆
Δ
X
for
the
corresponding
open
subgroups.
Observe
that
our
assumption
that
the
restricted
map
D
x
→
Q
is
trivial
implies
that
every
cusp
of
X
log
is
K-rational.
Let
us
write
ι
(respectively,
ι)
for
the
automorphism
of
X
log
(respectively,
X
log
)
determined
by
“multiplication
by
−1”
on
the
underlying
elliptic
curve
relative
to
choosing
the
unique
cusp
of
X
log
(respectively,
relative
to
some
choice
of
a
cusp
of
X
log
)
as
the
origin.
Thus,
if
we
denote
the
stack-theoretic
quotient
of
X
log
(respectively,
X
log
)
by
the
action
of
ι
(respectively,
ι)
by
C
log
(respectively,
C
log
),
then
we
have
a
cartesian
commutative
diagram:
X
log
⏐
⏐
−→
X
log
⏐
⏐
C
log
−→
C
log
We
shall
write
Π
C
,
Π
C
for
the
respective
(profinite)
étale
fundamental
groups
of
C
log
,
C
log
.
Thus,
we
obtain
subgroups
Δ
C
⊆
Π
C
,
Δ
C
⊆
Π
C
[i.e.,
the
kernels
of
the
natural
surjections
to
G
K
];
moreover,
by
forming
the
quotient
by
the
kernels
of
the
quotients
Π
X
Π
X
,
Π
X
Π
X
,
we
obtain
quotients
Π
C
Π
C
,
Π
C
ell
Π
C
,
Δ
C
Δ
C
,
Δ
C
Δ
C
.
Similarly,
the
quotient
Δ
X
Δ
X
determines
a
ell
quotient
Δ
C
Δ
C
.
Let
of
Gal(X/C)
∼
=
Z/2Z.
ι
∈
Δ
C
be
an
element
that
lifts
the
nontrivial
element
Definition
2.1.
We
shall
refer
to
a
smooth
log
orbicurve
over
K
that
arises,
up
to
isomorphism,
as
the
smooth
log
orbicurve
X
log
(respectively,
C
log
)
con-
ell
structed
above
for
some
choice
of
Π
X
Q
as
being
of
type
(1,
l-tors)
(respectively,
(1,
l-tors)
±
).
We
shall
also
apply
this
terminology
to
the
associated
hyperbolic
orbicurves.
Remark
2.1.1.
Note
that
although
X
log
→
X
log
is
[by
construction]
Galois,
with
Gal(X/X)
∼
=
Q,
the
covering
C
log
→
C
log
fails
to
be
Galois
in
general.
More
precisely,
no
nontrivial
automorphism
∈
Gal(X/X)
of,
say,
odd
order
descends
to
an
automorphism
of
C
log
over
C
log
.
Indeed,
this
follows
from
the
fact
that
ι
acts
on
Q
by
multiplication
by
−1.
34
Proposition
2.2.
Then:
SHINICHI
MOCHIZUKI
(The
Inversion
Automorphism)
Suppose
that
l
is
odd.
(i)
The
conjugation
action
of
ι
on
the
rank
two
(Z/lZ)-module
Δ
X
deter-
mines
a
direct
product
decomposition
ell
Δ
X
∼
=
Δ
X
×
Δ
Θ
into
eigenspaces,
with
eigenvalues
−1
and
1,
respectively,
that
is
compatible
with
the
conjugation
action
of
Π
X
.
Denote
by
ell
s
ι
:
Δ
X
→
Δ
X
ell
the
resulting
splitting
of
the
natural
surjection
Δ
X
Δ
X
.
(ii)
In
the
notation
of
(i),
the
normal
subgroup
Im(s
ι
)
⊆
Π
X
induces
an
iso-
morphism
∼
D
x
→
Π
X
/Im(s
ι
)
over
G
K
.
In
particular,
any
section
of
the
H
1
(G
K
,
Δ
Θ
)
∼
=
K
×
/(K
×
)
l
-torsor
of
splittings
of
D
x
G
K
determines
a
covering
X
log
→
X
log
whose
corresponding
open
subgroup
we
denote
by
Π
X
⊆
Π
X
.
Here,
the
“geometric
ell
portion”
Δ
X
of
Π
X
maps
isomorphically
onto
Δ
X
[hence
is
a
cyclic
group
of
order
l],
i.e.,
we
have
Δ
X
=
Im(s
ι
),
Δ
X
=
Δ
X
·
Δ
Θ
.
Finally,
the
image
of
ι
in
Δ
C
/Δ
X
may
be
characterized
as
the
unique
coset
of
Δ
C
/Δ
X
that
lifts
the
nontrivial
element
of
Gal(X/C)
=
Δ
C
/Δ
X
and
normalizes
the
subgroup
Δ
X
⊆
Δ
C
.
(iii)
There
exists
a
unique
coset
∈
Δ
C
/Δ
X
such
that
ι
has
order
2
if
and
only
if
it
belongs
to
this
coset.
If
we
choose
ι
to
have
order
2,
then
the
open
subgroup
generated
by
Π
X
and
ι
in
Π
C
[or,
alternatively,
the
open
subgroup
generated
by
G
K
∼
=
Π
X
/Δ
X
and
ι
in
Π
C
/Δ
X
]
determines
a
double
covering
X
log
→
C
log
which
fits
into
a
cartesian
commutative
diagram
X
log
⏐
⏐
−→
X
log
⏐
⏐
C
log
−→
C
log
—
where
X
log
is
as
in
(ii).
Proof.
Assertions
(i)
and
(ii)
are
immediate
from
the
definitions.
To
verify
asser-
tion
(iii),
we
observe
that
D
x
∼
=
Π
X
/Δ
X
is
of
index
2
in
Π
C
/Δ
X
.
Thus,
ι
normal-
izes
D
x
∼
=
Π
X
/Δ
X
.
Since,
moreover,
l
is
odd
[so
H
1
(G
K
,
Δ
Θ
),
Δ
Θ
/H
0
(G
K
,
Δ
Θ
)
THE
ÉTALE
THETA
FUNCTION
35
have
no
elements
of
order
2],
and
conjugation
by
ι
induces
the
identity
on
Δ
Θ
and
G
K
,
it
follows
that
ι
centralizes
D
x
∼
=
Π
X
/Δ
X
,
hence
[a
fortiori]
G
K
∼
=
Π
X
/Δ
X
.
Now
assertion
(iii)
follows
immediately.
Remark
2.2.1.
We
shall
not
discuss
the
case
of
even
l
in
detail
here.
Nevertheless,
we
pause
briefly
to
observe
that
if
l
=
2,
then
[since
Δ
Θ
lies
in
the
center
of
Δ
X
]
the
ell
automorphism
±
∈
Gal(
Ẍ/C)
∼
=
Δ
C
of
§1
acts
naturally
on
the
exact
sequence
ell
ell
1
→
Δ
Θ
→
Δ
X
→
Δ
X
→
1.
Since
this
action
is
clearly
trivial
on
Δ
Θ
,
Δ
X
,
one
ell
verifies
immediately
that
this
action
determines
a
homomorphism
Δ
X
→
Δ
Θ
,
i.e.,
in
effect,
a
2-torsion
point
[so
long
as
the
homomorphism
is
nontrivial]
of
the
elliptic
curve
underlying
X
log
.
Thus,
by
considering
the
case
where
K
is
the
field
of
moduli
of
this
elliptic
curve
[so
that
G
K
permutes
the
2-torsion
points
transitively],
we
ell
conclude
that
this
homomorphism
must
be
trivial,
i.e.,
that
every
element
of
Δ
X
admits
an
±
-invariant
lifting
to
Δ
X
.
Definition
2.3.
We
shall
refer
to
a
smooth
log
orbicurve
over
K
that
arises,
up
to
isomorphism,
as
the
smooth
log
orbicurve
X
log
(respectively,
C
log
)
constructed
in
Proposition
2.2
above
as
being
of
type
(1,
l-tors
Θ
)
(respectively,
(1,
l-tors
Θ
)
±
).
We
shall
also
apply
this
terminology
to
the
associated
hyperbolic
orbicurves.
Remark
2.3.1.
Thus,
one
may
think
of
the
“single
underline”
in
the
notation
X
log
,
C
log
as
denoting
the
result
of
“extracting
a
single
copy
of
Z/lZ”,
and
the
“double
underline”
in
the
notation
X
log
,
C
log
as
denoting
the
result
of
“extracting
two
copies
of
Z/lZ”.
Proposition
2.4.
(Characteristic
Nature
of
Coverings)
For
=
α,
β,
let
X
log
be
a
smooth
log
curve
of
type
(1,
l-tors
Θ
)
over
a
finite
extension
K
log
log
log
of
Q
p
,
where
l
is
odd;
write
C
log
,
X
log
,
C
,
X
,
C
for
the
related
smooth
log
has
stable
log
orbicurves
[as
in
the
above
discussion].
Assume
further
that
X
reduction
over
O
K
,
with
singular
and
split
special
fiber.
Then
any
isomorphism
of
topological
groups
∼
∼
tp
tp
tp
γ
:
Π
tp
X
→
Π
X
(respectively,
γ
:
Π
X
→
Π
X
;
α
β
γ
:
Π
tp
C
α
α
β
∼
tp
∼
tp
→
Π
tp
C
;
γ
:
Π
C
α
→
Π
C
β
)
β
induces
isomorphisms
compatible
with
the
various
natural
maps
between
the
respec-
log
log
log
log
log
log
log
(respectively,
X
,
C
,
Ÿ
log
;
tive
“Π
tp
’s”
of
X
log
,
X
,
C
,
C
,
C
,
Ÿ
log
log
log
log
log
log
log
log
log
C
log
,
C
,
X
,
X
,
X
,
Ÿ
;
C
,
X
,
X
,
Ÿ
)
where
=
α,
β.
Proof.
As
in
the
proof
of
Proposition
1.8,
it
follows
from
our
assumption
that
log
the
hyperbolic
curve
determined
by
X
admits
a
K
-core
that
γ
induces
an
iso-
tp
∼
tp
morphism
Π
C
α
→
Π
C
β
[cf.
[Mzk3],
Theorem
2.4]
which
[cf.
[Mzk2],
Lemma
1.3.8]
36
SHINICHI
MOCHIZUKI
∼
tp
induces
an
isomorphism
Δ
tp
C
α
→
Δ
C
β
;
moreover,
this
last
isomorphism
induces
[by
considering
open
subgroups
of
index
2
whose
profinite
completions
contain
no
∼
tp
torsion
elements]
an
isomorphism
Δ
tp
X
α
→
Δ
X
β
,
hence
also
[by
considering
the
con-
tp
jugation
action
of
Π
tp
C
on
an
appropriate
abelian
quotient
of
Δ
X
as
in
the
proof
∼
tp
of
Proposition
1.8]
an
isomorphism
Π
tp
X
α
→
Π
X
β
,
which
preserves
the
decomposi-
tion
groups
of
cusps
[cf.
[Mzk14],
Theorem
6.5,
(iii)].
Also,
by
the
definition
of
∼
∼
tp
Δ
X
,
the
isomorphism
Δ
tp
X
α
→
Δ
X
β
determines
an
isomorphism
Δ
X
α
→
Δ
X
β
.
In
light
of
these
observations,
the
various
assertions
of
Proposition
2.4
follow
immedi-
ately
from
the
definitions
[cf.
also
Proposition
2.2;
Theorem
1.6,
(i);
the
proof
of
Proposition
1.8].
Now,
we
return
to
the
discussion
of
§1.
In
particular,
we
assume
that
K
is
a
finite
extension
of
Q
p
.
Definition
2.5.
Suppose
that
l
and
the
residue
characteristic
of
K
are
odd,
and
that
K
=
K̈
[cf.
Definition
1.7
and
the
preceding
discussion].
ell
(i)
Suppose,
in
the
situation
of
Definitions
2.1,
2.3,
that
the
quotient
Π
X
Q
determined
by
the
quotient
Π
tp
Z
factors
through
the
natural
quotient
Π
X
Z
X
discussed
at
the
beginning
of
§1,
and
that
the
choice
of
a
splitting
of
D
x
→
G
K
[cf.
Proposition
2.2,
(ii)]
that
determined
the
covering
X
log
→
X
log
is
compatible
with
the
“{±1}-structure”
of
Theorem
1.10,
(iii).
Then
we
shall
say
that
the
orbicurve
of
type
(1,
l-tors)
(respectively,
(1,
l-tors
Θ
);
(1,
l-tors)
±
;
(1,
l-tors
Θ
)
±
)
under
consid-
eration
is
of
type
(1,
Z/lZ)
(respectively,
(1,
(Z/lZ)
Θ
);
(1,
Z/lZ)
±
;
(1,
(Z/lZ)
Θ
)
±
).
(ii)
In
the
notation
of
the
above
discussion
and
the
discussion
at
the
end
of
§1,
we
shall
refer
to
a
smooth
log
orbicurve
isomorphic
to
the
smooth
log
orbicurve
Ẋ
log
(respectively,
Ẋ
log
;
Ċ
log
;
Ċ
log
)
obtained
by
taking
the
composite
of
the
covering
X
log
(respectively,
X
log
;
C
log
;
C
log
)
of
C
log
with
the
covering
Ċ
log
→
C
log
,
as
being
of
type
(1,
μ
2
×
Z/lZ)
(respectively,
(1,
μ
2
×
(Z/lZ)
Θ
);
(1,
μ
2
×
Z/lZ)
±
;
(1,
μ
2
×
(Z/lZ)
Θ
)
±
).
Remark
2.5.1.
Thus,
the
irreducible
components
of
the
special
fiber
of
C,
Ċ
may
be
naturally
identified
with
the
elements
of
(Z/lZ)/{±1}
—
cf.
Corollary
2.9
below
for
more
details.
Proposition
2.6.
(Characteristic
Nature
of
Coverings)
For
=
α,
β,
let
us
assume
that
we
have
smooth
log
orbicurves
as
in
the
above
discussion,
over
a
finite
extension
K
of
Q
p
.
Then
any
isomorphism
of
topological
groups
∼
∼
γ
:
Π
tp
→
Π
tp
(respectively,
γ
:
Π
tp
→
Π
tp
;
Ẋ
Ẋ
Ẋ
Ẋ
α
β
γ
:
Π
tp
Ċ
α
α
∼
∼
→
Π
tp
;
γ
:
Π
tp
→
Π
tp
)
Ċ
Ċ
α
Ċ
β
β
β
THE
ÉTALE
THETA
FUNCTION
37
induces
isomorphisms
compatible
with
the
various
natural
maps
between
the
respec-
log
log
log
(respectively,
X
log
tive
“Π
tp
’s”
of
X
log
;
C
;
C
)
and
Ċ
,
where
=
α,
β.
A
similar
statement
holds
when
“Π
tp
”
is
replaced
by
“Π”.
Proof.
The
proof
is
entirely
similar
to
the
proofs
of
Propositions
1.8,
2.4.
Remark
2.6.1.
Suppose,
for
simplicity,
that
K
contains
a
primitive
l-th
root
of
unity.
Then
we
observe
in
passing
that
by
applying
the
Propositions
2.4,
2.6
to
“isomorphisms
of
fundamental
groups
arising
from
isomorphisms
of
the
orbi-
curves
in
question”
[cf.
also
Remark
2.1.1],
one
computes
easily
that
the
groups
of
K-linear
automorphisms
“Aut
K
(−)”
of
the
various
smooth
log
orbicurves
under
consideration
are
given
as
follows:
Aut
K
(X
log
)
=
μ
l
×
{±1};
Aut
K
(X
log
)
=
Z/lZ
{±1}
Aut
K
(C
log
)
=
μ
l
;
Aut
K
(C
log
)
=
{1}
—
where
μ
l
denotes
the
group
of
l-th
roots
of
unity
in
K,
and
the
semi-direct
product
“”
is
with
respect
to
the
natural
multiplicative
action
of
±1
on
Z/lZ;
the
“Aut
K
(−)’s”
of
the
various
“once-dotted
versions”
of
these
orbicurves
[cf.
Defini-
tion
2.5,
(ii)]
are
given
by
taken
the
direct
product
of
the
“Aut
K
(−)’s”
listed
above
with
Gal(
Ċ
log
/C
log
)
∼
=
{±1}.
Next,
we
consider
étale
theta
functions.
First,
let
us
observe
that
the
covering
log
Ÿ
log
→
C
log
factors
naturally
through
Ẋ
.
Thus,
the
class
[which
is
only
well-
×
defined
up
to
a
O
K
-multiple]
η̈
Θ
∈
H
1
(Π
tp
,
Δ
Θ
)
Ÿ
∼
/Π
tp
of
§1
—
as
well
as
the
corresponding
Π
tp
=
Z-orbit
η̈
Θ,Z
—
may
be
thought
of
Ẋ
Ÿ
log
log
as
objects
associated
to
the
“Π
tp
”
of
Ẋ
,
Ċ
,
X
log
,
C
log
.
On
the
other
hand,
the
composites
of
the
coverings
Ÿ
log
→
C
log
,
Y
log
→
C
log
with
C
log
→
C
log
determine
new
coverings
log
Ÿ
→
Ÿ
log
;
Y
log
→
Y
log
of
degree
l.
Moreover,
the
choice
of
a
splitting
of
D
x
→
G
K
[cf.
Proposition
2.2,
(ii)]
that
determined
the
covering
X
log
→
X
log
determines
[by
considering
→
(Π
tp
)
Θ
—
cf.
Proposition
1.5,
(ii)]
a
specific
class
the
natural
map
D
x
→
Π
tp
Ÿ
Ÿ
×
l
∈
H
1
(Π
tp
,
Δ
Θ
⊗
Z/lZ),
which
may
be
thought
of
as
a
choice
of
η̈
Θ
up
to
an
(O
K
)
-
Ÿ
×
multiple
[i.e.,
as
opposed
to
only
up
to
a
O
K
-multiple].
Now
it
is
a
tautology
that,
log
→
Ÿ
log
[which
was
determined,
in
effect,
by
upon
restriction
to
the
covering
Ÿ
the
choice
of
a
splitting
of
D
x
→
G
K
],
the
class
η̈
Θ
determines
a
class
η̈
Θ
∈
H
1
(Π
tp
,
l
·
Δ
Θ
)
Ÿ
38
SHINICHI
MOCHIZUKI
tp
tp
∼
—
as
well
as
a
corresponding
Π
tp
/Π
tp
=
Π
Ẋ
/Π
Ÿ
∼
=
l
·
Z-orbit
η̈
Θ,l·Z
—
which
may
Ẋ
Ÿ
be
thought
of
as
objects
associated
to
the
“Π
tp
”
of
Ẋ
which
satisfy
the
following
property:
H
1
(Π
tp
,
l
·
Δ
Θ
)
Ÿ
log
,
Ċ
log
,
X
log
,
C
log
,
and
η̈
Θ
→
η̈
Θ
|
Ÿ
∈
H
1
(Π
tp
,
Δ
Θ
)
Ÿ
[relative
to
the
natural
inclusion
l
·
Δ
Θ
→
Δ
Θ
].
That
is
to
say,
at
a
more
intuitive
level,
η̈
Θ
may
be
thought
of
as
an
“l-th
root
of
the
étale
theta
function”.
In
the
following,
we
shall
also
consider
the
l
·
Z-orbit
η̈
Θ,l·Z
of
η̈
Θ
,
as
well
as
the
Π
tp
/Π
tp
∼
=
X
tp
∼
Π
tp
X
/Π
Ÿ
=
{(l
·
Z)
×
μ
2
}-orbits
η̈
Θ,l·Z×μ
2
,
Ÿ
η̈
Θ,l·Z×μ
2
tp
∼
Θ,Z×μ
2
of
η̈
Θ
,
η̈
Θ
,
and
the
Π
tp
of
η̈
Θ
.
X
/Π
Ÿ
=
(Z
×
μ
2
)-orbit
η̈
Definition
2.7.
If
η̈
Θ,Z
is
of
standard
type,
then
we
shall
also
refer
to
η̈
Θ,l·Z
,
Θ,l·Z
Θ,l·Z×μ
2
Θ,l·Z×μ
2
η̈
,
η̈
,
η̈
,
η̈
Θ,Z×μ
2
as
being
of
standard
type.
Corollary
2.8.
(Constant
Multiple
Rigidity
of
Roots
of
the
Étale
Theta
Function)
For
=
α,
β,
let
us
assume
that
we
have
smooth
log
orbicurves
as
in
the
above
discussion,
over
a
finite
extension
K
of
Q
p
.
Let
∼
∼
tp
tp
tp
γ
:
Π
tp
X
→
Π
X
(respectively,
γ
:
Π
X
→
Π
X
;
α
β
γ
:
Π
tp
C
α
α
β
∼
tp
∼
tp
→
Π
tp
C
;
γ
:
Π
C
α
→
Π
C
β
)
β
be
an
isomorphism
of
topological
groups.
Then:
(i)
The
isomorphism
γ
preserves
the
property
[cf.
Theorem
1.6,
(iii)]
that
Θ,Z×μ
2
Θ,l·Z×μ
2
(respectively,
η̈
;
η̈
Θ,l·Z×μ
2
;
η̈
)
be
of
standard
type
—
a
η̈
property
that
determines
this
collection
of
classes
up
to
multiplication
by
a
root
of
unity
of
order
l
(respectively,
1;
l;
1).
Θ,l·Z×μ
2
(ii)
Suppose
further
that
the
cusps
of
X
are
rational
over
K
,
that
the
residue
characteristic
of
K
is
prime
to
l,
and
that
K
contains
a
primitive
l-th
root
of
unity.
Then
the
{±1}-[i.e.,
μ
2
-]
structure
of
Theorem
1.10,
(iii),
deter-
mines
a
μ
2l
(respectively,
μ
2
;
μ
2l
;
μ
2
)-structure
[cf.
[Mzk13],
Corollary
4.12]
on
log
log
×
∧
)
-torsor
at
the
cusps
of
X
log
(respectively,
X
log
the
(K
;
X
;
X
).
Moreover,
this
μ
2l
(respectively,
μ
2
;
μ
2l
;
μ
2
)-structure
is
compatible
with
the
canonical
in-
tegral
structure
[cf.
[Mzk13],
Definition
4.1,
(iii)]
determined
by
the
stable
model
log
and
preserved
by
γ.
of
X
(iii)
If
the
data
for
=
α,
β
are
equal,
and
γ
arises
[cf.
Proposition
2.6]
(respectively,
Π
tp
;
Π
tp
;
Π
tp
),
then
from
an
inner
automorphism
of
Π
tp
Ẋ
Ẋ
Ċ
Ċ
THE
ÉTALE
THETA
FUNCTION
39
γ
preserves
η̈
Θ,l·Z
(respectively,
η̈
Θ,l·Z
;
η̈
Θ,l·Z
;
η̈
Θ,l·Z
)
[i.e.,
without
any
constant
multiple
indeterminacy].
Proof.
First,
let
us
recall
the
characteristic
nature
of
the
various
coverings
involved
[cf.
Propositions
2.4,
2.6].
Now
assertion
(i)
follows
immediately
from
Theorem
1.10,
(i),
and
the
definitions;
assertion
(ii)
follows
immediately
from
Theorem
1.10,
(iii),
and
the
definitions;
assertion
(iii)
follows
immediately
from
Remark
1.9.1.
Before
proceeding,
we
pause
to
take
a
closer
look
at
the
cusps
of
the
various
smooth
log
orbicurves
under
consideration.
First,
we
recall
from
the
discussion
preceding
Lemma
1.2
that
the
irreducible
components
of
the
special
fiber
of
Y
log
may
be
assigned
labels
∈
Z,
in
a
natural
fashion.
These
labels
thus
determine
labels
∈
Z
for
the
cusps
of
Y
log
[i.e.,
by
considering
the
irreducible
component
of
the
special
fiber
of
Y
log
that
contains
the
closure
in
Y
of
the
cusp
in
question].
Moreover,
by
considering
the
covering
Ÿ
the
cusps
of
Ÿ
log
Ẋ
log
log
→
Y
log
,
we
thus
obtain
labels
∈
Z
for
.
Since
the
various
smooth
log
orbicurves
;
Ẋ
log
;
Ċ
log
;
Ċ
log
;
X
log
;
X
log
;
C
log
;
C
log
log
all
appear
as
subcoverings
of
the
covering
Ÿ
→
X
log
,
we
thus
obtain
labels
∈
Z
for
the
cusps
of
these
smooth
log
orbicurves,
which
are
well-defined
up
to
a
certain
indeterminacy.
If
we
write
(Z/lZ)
±
for
the
quotient
of
the
set
Z/lZ
by
the
natural
multiplicative
action
of
±1,
then
it
follows
immediately
from
the
construction
of
these
smooth
log
orbicurves
that
this
indeterminacy
is
such
that
the
labels
for
the
cusps
of
these
smooth
log
orbicurves
may
be
thought
of
as
well-defined
elements
of
(Z/lZ)
±
.
Corollary
2.9.
(Labels
of
Cusps)
Suppose
that
K
contains
a
primitive
l-th
root
of
unity.
Then
for
each
of
the
smooth
log
orbicurves
Ẋ
log
;
Ċ
log
;
Ċ
log
;
X
log
;
C
log
;
C
log
[as
defined
in
the
above
discussion],
the
labels
of
the
above
discussion
determine
a
bijection
of
the
set
(Z/lZ)
±
with
the
set
of
“Aut
K
(−)”-orbits
[cf.
Remark
2.6.1]
of
the
cusps
of
the
smooth
log
orbicurve.
Moreover,
in
the
case
of
X
log
,
C
log
,
and
C
log
,
these
bijections
are
preserved
by
arbitrary
isomorphisms
of
topological
groups
“γ”
as
in
Corollary
2.8.
Proof.
The
asserted
bijections
follow
immediately
by
tracing
through
the
def-
initions
of
the
various
smooth
log
orbicurves
[cf.
Remark
2.6.1].
With
regard
40
SHINICHI
MOCHIZUKI
to
showing
that
these
bijections
are
preserved
by
“γ”
as
in
Corollary
2.8,
we
re-
duce
immediately
by
Proposition
2.4
to
the
case
of
C
log
;
in
this
case,
the
desired
preservation
follows
immediately
from
the
definition
of
the
labels
in
question
in
the
discussion
above,
together
with
the
fact
that
such
γ
always
preserve
the
dual
graphs
of
the
special
fibers
of
the
orbicurves
in
question
[cf.
[Mzk2],
Lemma
2.3].
Remark
2.9.1.
We
observe
in
passing
that
a
bijection
as
in
Corollary
2.9
fails
to
log
hold
for
Ẋ
,
X
log
—
cf.
Remark
2.6.1.
Remark
2.9.2.
In
the
situation
of
Corollary
2.8,
(ii),
we
make
the
following
observation,
relative
to
the
labels
of
Corollary
2.9:
The
2l
(respectively,
2)
trivial-
×
∧
izations
of
the
(K
)
-torsor
at
a
cusp
labeled
0
(respectively,
an
arbitrary
cusp)
of
log
X
(respectively,
X
log
)
determined
by
the
μ
2l
(respectively,
μ
2
)-structure
under
discussion
are
permuted
transitively
by
the
subgroup
of
Aut
K
(X
log
)
(respectively,
Aut
K
(X
log
))
[cf.
Remarks
2.1.1,
2.6.1,
2.9.1]
that
stabilizes
the
cusp.
In
the
case
of
log
X
,
at
cusps
with
nonzero
labels,
the
subgroup
of
the
corresponding
“Aut
K
(−)”
that
stabilizes
the
cusp
permutes
the
2l
trivializations
under
consideration
via
the
action
of
μ
l
[hence
has
precisely
two
orbits].
Next,
let
N
≥
1
be
an
integer;
set
def
Δ
μ
N
=
(Z/N
Z(1));
def
Π
μ
N
,K
=
Δ
μ
N
G
K
—
so
we
have
a
natural
exact
sequence
1
→
Δ
μ
N
→
Π
μ
N
,K
→
G
K
→
1.
Definition
2.10.
If
Π
G
K
is
a
topological
group
equipped
with
an
augmenta-
tion
[i.e.,
a
surjection]
to
G
K
,
then
we
shall
write
def
Π[μ
N
]
=
Π
×
G
K
Π
μ
N
,K
and
refer
to
Π[μ
N
]
as
the
cyclotomic
envelope
of
Π
G
K
[or
Π,
for
short].
Also,
def
if
Δ
=
Ker(Π
G
K
),
then
we
shall
write
def
Δ[μ
N
]
=
Ker(Π[μ
N
]
G
K
)
—
so
Δ[μ
N
]
=
Δ×Δ
μ
N
;
we
have
a
natural
exact
sequence
1
→
Δ[μ
N
]
→
Π[μ
N
]
→
G
K
→
1.
Note
that,
by
construction,
we
have
a
tautological
section
G
K
→
Π
μ
N
,K
of
Π
μ
N
,K
G
K
,
which
determines
a
section
s
alg
Π
:
Π
→
Π[μ
N
]
of
Π[μ
N
]
Π,
which
we
shall
also
call
tautological.
We
shall
refer
to
a
μ
N
-orbit,
relative
to
the
action
of
μ
N
by
conjugation,
of
objects
associated
to
Π[μ
N
]
[e.g.,
THE
ÉTALE
THETA
FUNCTION
41
subgroups
of
Π[μ
N
],
homomorphisms
from
Π[μ
N
]
to
another
topological
group,
etc.]
as
a
μ
N
-conjugacy
class.
Proposition
2.11.
(General
Properties
of
the
Cyclotomic
Envelope)
For
=
α,
β,
let
Π
G
K
be
an
open
subgroup
of
either
the
tempered
or
the
profinite
fundamental
group
of
a
hyperbolic
orbicurve
over
a
finite
extension
K
of
Q
p
;
write
Δ
for
the
kernel
of
the
natural
morphism
Π
→
G
K
.
Then:
(i)
The
kernel
of
the
natural
surjection
Δ
[μ
N
]
Δ
is
equal
to
the
center
of
∼
Δ
[μ
N
].
In
particular,
any
isomorphism
of
topological
groups
Δ
α
[μ
N
]
→
Δ
β
[μ
N
]
is
compatible
with
the
natural
surjections
Δ
[μ
N
]
Δ
.
(ii)
The
kernel
of
the
natural
surjection
Π
[μ
N
]
Π
is
equal
to
the
union
of
the
centralizers
of
the
open
subgroups
of
Π
[μ
N
].
In
particular,
any
isomorphism
∼
of
topological
groups
Π
α
[μ
N
]
→
Π
β
[μ
N
]
is
compatible
with
the
natural
surjections
Π
[μ
N
]
Π
.
Proof.
Assertions
(i),
(ii)
follow
immediately
from
the
“temp-slimness”
[i.e.,
the
triviality
of
the
centralizers
of
all
open
subgroups
of]
Δ
,
Π
[cf.
[Mzk14],
Example
3.10].
Next,
let
us
write
def
Θ
Π
Θ
C
=
Π
C
/Ker(Δ
X
Δ
X
);
def
Θ
Δ
Θ
C
=
Δ
C
/Ker(Δ
X
Δ
X
)
and,
in
a
similar
vein,
denote
by
means
of
a
superscript
“Θ”
the
quotients
of
the
log
log
tempered
and
profinite
fundamental
groups
of
Ẋ
,
Ẋ
,
Ẋ
log
,
X
log
,
X
log
,
X
log
,
Ċ
log
,
Ċ
log
,
Ċ
log
,
C
log
,
C
log
,
C
log
,
determined
by
these
quotients.
Also,
let
us
write
def
ell
Π
ell
C
=
Π
C
/Ker(Δ
X
Δ
X
);
def
ell
Δ
ell
C
=
Δ
C
/Ker(Δ
X
Δ
X
)
and
denote
by
means
of
a
superscript
“ell”
the
various
induced
quotients.
Proposition
2.12.
(The
Cyclotomic
Envelope
of
the
Theta
Quotient)
Let
Δ
∗
be
one
of
the
following
topological
groups:
Δ
tp
X
;
Δ
tp
;
Ċ
Δ
tp
C
;
Δ
X
;
Δ
Ċ
;
Δ
C
Then:
(i)
We
have
an
inclusion
ell
Θ
=
l
·
Δ
Θ
⊆
Δ
Θ
Ker
Δ
Θ
∗
Δ
∗
∗
,
Δ
∗
of
subgroups
of
Δ
Θ
∗
.
42
SHINICHI
MOCHIZUKI
(ii)
The
intersection
Θ
Δ
Θ
∗
[μ
N
],
Δ
∗
[μ
N
]
(l
·
Δ
Θ
)[μ
N
]
⊆
(l
·
Δ
Θ
)[μ
N
]
⊆
Δ
Θ
∗
[μ
N
]
coincides
with
the
image
of
the
restriction
of
the
tautological
section
of
Δ
Θ
∗
[μ
N
]
to
l
·
Δ
.
Δ
Θ
∗
Θ
Proof.
First,
we
consider
the
inclusion
of
assertion
(i).
Now
since
l
is
odd,
the
prime-to-2
portion
of
this
inclusion
then
follows
immediately
from
the
well-known
log
Θ
structure
of
the
“theta-group”
(Δ
tp
→
X
)
[cf.
also
the
definition
of
the
covering
C
log
log
log
C
];
in
the
case
of
X
,
C
,
the
pro-2
portion
of
this
inclusion
follows
similarly.
log
On
the
other
hand,
in
the
case
of
Ċ
,
the
pro-2
portion
of
this
inclusion
follows
from
the
fact
that,
in
the
notation
of
Remark
2.2.1
[i.e.,
more
precisely,
when
“l
=
Δ
Δ
2”],
if
we
denote
by
Δ
Z
,
μ
∈
Δ
X
(⊆
Δ
C
),
±
∈
Δ
C
liftings
to
Δ
C
of
the
elements
of
ell
Δ
C
determined
by
the
automorphisms
“
Z
”,
“
μ
”,
“
±
”
of
the
discussion
preceding
Δ
Δ
Definition
1.7,
then
Δ
±
commutes
with
Z
,
μ
[cf.
the
observation
of
Remark
2.2.1],
so
the
commutator
Δ
Δ
Δ
Δ
[
Δ
Z
,
±
·
μ
]=[
Z
,
μ
]
is
a
nonzero
element
of
Δ
Θ
.
Assertion
(ii)
follows
formally
from
assertion
(i).
Remark
2.12.1.
Note
that
the
inclusion
of
Proposition
2.12,
(i)
—
which
will
be
crucial
in
the
theory
to
follow
—
fails
to
hold
if
one
replaces
X
log
,
Ċ
Ẋ
log
,
Ẋ
log
[one
has
problems
at
the
prime
2];
Ẋ
2
and
the
primes
dividing
l];
X
log
,
Ċ
log
log
log
,
C
log
by
[one
has
problems
at
the
prime
,
C
log
[one
has
problems
at
the
primes
log
dividing
l].
(There
is
no
problem,
however,
if
one
replaces
X
log
,
Ċ
,
C
log
by
X
log
,
Ċ
log
,
C
log
since
this
just
corresponds
to
the
case
l
=
1.)
Indeed,
the
original
motivation
for
the
introduction
of
the
slightly
complicated
coverings
X
log
,
Ċ
C
log
was
precisely
to
avoid
these
problems.
log
,
Next,
let
us
observe
that,
by
subtracting
[i.e.,
if
we
treat
cohomology
classes
additively]
the
reduction
modulo
N
of
any
member
of
the
collection
of
[cocycles
determined
by
the
collection
of]
classes
η̈
Θ,l·Z×μ
2
in
H
1
(Π
tp
,
l
·
Δ
Θ
)
from
the
[com-
Ÿ
posite
with
the
inclusion
into
Π
tp
Y
[μ
N
]
of
the]
tautological
section
def
s
alg
=
s
alg
:
Π
tp
→
Π
tp
[μ
N
]
→
Π
tp
Y
[μ
N
]
Ÿ
Π
tp
Ÿ
Ÿ
Ÿ
—
where
we
apply
the
natural
isomorphism
μ
N
∼
=
(l
·
Δ
Θ
)
⊗
(Z/N
Z)
—
yields
a
tp
tp
Θ
new
homomorphism:
s
Ÿ
:
Π
Ÿ
→
Π
Y
[μ
N
].
THE
ÉTALE
THETA
FUNCTION
43
Now
since
the
tautological
section
s
alg
extends
to
a
tautological
section
s
alg
:
Ÿ
Π
tp
C
tp
tp
tp
Π
tp
C
→
Π
C
[μ
N
]
[where
we
regard
Π
Y
[μ
N
]
as
a
subgroup
of
Π
C
[μ
N
]],
it
follows
that
the
natural
outer
action
tp
tp
tp
tp
tp
Gal(Y
/C)
∼
=
Π
C
/Π
Y
∼
=
Π
C
[μ
N
]/Π
Y
[μ
N
]
→
Out(Π
Y
[μ
N
])
alg
of
Gal(Y
/C)
on
Π
tp
Y
[μ
N
]
fixes
the
image
of
s
Ÿ
,
up
to
conjugation
by
an
element
of
μ
N
.
In
particular,
it
follows
immediately
from
the
definitions
that
the
various
s
Θ
Ÿ
that
arise
from
different
choices
of
[a
cocycle
contained
in]
a
class
∈
η̈
Θ,l·Z×μ
2
are
obtained
as
Π
tp
X
[μ
N
]-conjugates
[where
we
recall
that
we
have
a
natural
isomorphism
∼
Θ
(Π
tp
X
)
Gal(
Ÿ
/X)
→
(l·Z)×μ
2
]
of
any
given
s
Ÿ
.
[Here,
we
note
that
“conjugation
by
an
element
of
μ
N
”
corresponds
precisely
to
modifying
a
cocycle
by
a
coboundary.]
Note,
moreover,
that
we
have
a
natural
outer
action
∼
tp
K
×
(K
×
)/(K
×
)
N
→
H
1
(G
K
,
μ
N
)
→
H
1
(Π
tp
Y
,
μ
N
)
→
Out(Π
Y
[μ
N
])
∼
—
where
the
“
→
”
is
the
Kummer
map
—
of
K
×
on
Π
tp
Y
[μ
N
],
which
induces
tp
the
trivial
outer
action
on
both
the
quotient
Π
tp
Y
[μ
N
]
Π
Y
and
the
kernel
of
×
-
this
quotient.
Relative
to
this
natural
outer
action,
replacing
η̈
Θ,l·Z×μ
2
by
an
O
K
×
by
an
O
K
-
multiple
of
η̈
Θ,l·Z×μ
2
[cf.
Proposition
1.3]
corresponds
to
replacing
s
Θ
Ÿ
.
conjugate
of
s
Θ
Ÿ
Definition
2.13.
In
the
notation
of
the
above
discussion:
(i)
Write
D
Y
⊆
Out(Π
tp
Y
[μ
N
])
×
∼
for
the
subgroup
of
Out(Π
tp
Y
[μ
N
])
generated
by
the
image
of
K
,
Gal(Y
/X)
(
=
l·Z).
:
Π
tp
→
Π
tp
We
shall
refer
to
s
Θ
Y
[μ
N
]
as
the
[mod
N
]
theta
section.
We
shall
refer
Ÿ
Ÿ
to
s
alg
:
Π
tp
→
Π
tp
Y
[μ
N
]
as
the
[mod
N
]
algebraic
section.
Ÿ
Ÿ
(ii)
We
shall
refer
to
as
a
[mod
N
]
model
mono-theta
environment
any
[ordered]
collection
of
data
as
folows:
(a)
the
topological
group
Π
tp
Y
[μ
N
];
(b)
the
subgroup
D
Y
⊆
Out(Π
tp
Y
[μ
N
]);
(c)
the
μ
N
-conjugacy
class
of
subgroups
⊆
Π
tp
Y
[μ
N
]
determined
by
the
image
.
of
the
theta
section
s
Θ
Ÿ
44
SHINICHI
MOCHIZUKI
We
shall
refer
to
as
a
[mod
N
]
mono-theta
environment
any
[ordered]
collection
of
data
consisting
of
a
topological
group
Π,
a
subgroup
D
Π
⊆
Out(Π),
and
a
collection
of
subgroups
s
Θ
Π
of
Π
such
that
there
exists
an
isomorphism
of
topological
groups
∼
tp
Π
→
Π
Y
[μ
N
]
[cf.
(a)]
mapping
D
Π
⊆
Out(Π)
to
D
Y
[cf.
(b)]
and
s
Θ
Π
to
the
μ
N
-conjugacy
class
of
(c).
[In
particular,
every
model
mono-theta
environment
determines
a
mono-theta
environment.]
We
shall
refer
to
as
an
isomorphism
of
∼
[mod
N
]
mono-theta
environments
M
→
M
between
two
[mod
N
]
mono-theta
environments
def
def
M
=
(Π
,
D
Π
,
s
Θ
M
=
(Π,
D
Π
,
s
Θ
Π
);
Π
)
∼
Θ
any
isomorphism
of
topological
groups
Π
→
Π
that
maps
D
Π
→
D
Π
,
s
Θ
Π
→
s
Π
.
If
N
|N
,
M
is
a
mod
N
mono-theta
environment,
and
M
is
a
mod
N
mono-theta
environment,
then
we
shall
refer
to
as
a
morphism
of
mono-theta
environments
∼
M
→
M
any
isomorphism
M
N
→
M
,
where
we
write
M
N
for
the
mod
N
mono-
theta
environment
induced
by
M.
(iii)
We
shall
refer
to
as
a
[mod
N
]
model
bi-theta
environment
any
[ordered]
collection
of
data
as
follows:
(a)
the
topological
group
Π
tp
Y
[μ
N
];
(b)
the
subgroup
D
Y
⊆
Out(Π
tp
Y
[μ
N
]);
(c)
the
μ
N
-conjugacy
class
of
subgroups
⊆
Π
tp
Y
[μ
N
]
determined
by
the
image
;
of
the
theta
section
s
Θ
Ÿ
(d)
the
μ
N
-conjugacy
class
of
subgroups
⊆
Π
tp
Y
[μ
N
]
determined
by
the
image
of
the
algebraic
section
s
alg
.
Ÿ
We
shall
refer
to
as
a
[mod
N
]
bi-theta
environment
any
collection
of
data
consisting
of
a
topological
group
Π,
a
subgroup
D
Π
⊆
Out(Π),
and
an
ordered
pair
of
collections
alg
of
subgroups
s
Θ
Π
,
s
Π
of
Π
such
that
there
exists
an
isomorphism
of
topological
∼
Θ
groups
Π
→
Π
tp
Y
[μ
N
]
[cf.
(a)]
mapping
D
Π
⊆
Out(Π)
to
D
Y
[cf.
(b)],
s
Π
to
the
μ
N
-conjugacy
class
of
(c),
and
s
alg
Π
to
the
μ
N
-conjugacy
class
of
(d).
[In
particular,
every
model
bi-theta
environment
determines
a
bi-theta
environment.]
We
shall
∼
refer
to
as
an
isomorphism
of
[mod
N
]
bi-theta
environments
B
→
B
between
two
[mod
N
]
bi-theta
environments
def
alg
B
=
(Π,
D
Π
,
s
Θ
Π
,
s
Π
);
alg
B
=
(Π
,
D
Π
,
s
Θ
Π
,
s
Π
)
def
∼
Θ
any
isomorphism
of
topological
groups
Π
→
Π
that
maps
D
Π
→
D
Π
,
s
Θ
Π
→
s
Π
,
alg
s
alg
Π
→
s
Π
.
If
N
|N
,
B
is
a
mod
N
bi-theta
environment,
and
B
is
a
mod
N
bi-
theta
environment,
then
we
shall
refer
to
as
a
morphism
of
bi-theta
environments
∼
B
→
B
any
isomorphism
B
N
→
B
,
where
we
write
B
N
for
the
mod
N
bi-theta
environment
induced
by
B.
THE
ÉTALE
THETA
FUNCTION
45
(iv)
In
the
situation
of
(iii),
if
η̈
Θ,l·Z×μ
2
is
of
standard
type,
then
we
shall
refer
to
the
resulting
[mod
N
]
model
bi-theta
environment
as
being
of
standard
type.
Proposition
2.14.
(Symmetries
of
Mono-theta
and
Bi-theta
Environ-
ments)
In
the
notation
of
the
above
discussion:
(i)
The
subset
(Δ
tp
)
Θ
[μ
N
]
⊇
(l
·
Δ
Θ
)[μ
N
]
Ÿ
tp
Θ
Θ
{γ(β)
·
β
−1
∈
(Δ
tp
Y
)
[μ
N
]
|
β
∈
(Δ
Y
)
[μ
N
],
γ
∈
Aut(Π
tp
Y
[μ
N
])
s.t.
the
image
of
γ
in
“Out”
belongs
to
D
Y
,
tp
and
γ
induces
the
identity
on
the
quotient
Π
tp
Y
[μ
N
]
Π
Y
G
K
}
coincides
with
the
image
of
the
tautological
section
of
(l
·
Δ
Θ
)[μ
N
]
(l
·
Δ
Θ
).
(ii)
Let
t
Θ
:
Π
tp
→
Π
tp
Y
[μ
N
]
Ÿ
Ÿ
,
relative
to
the
actions
of
K
×
,
(l
·
Z).
be
a
section
obtained
as
a
conjugate
of
s
Θ
Ÿ
with
coefficients
in
μ
N
obtained
by
subtracting
Write
δ
for
the
[1-]cocycle
of
Π
tp
Ÿ
from
t
Θ
and
s
Θ
Ÿ
Ÿ
[μ
N
])
α̈
δ
∈
Aut(Π
tp
Ÿ
[μ
N
]
obtained
by
“shifting”
for
the
automorphism
of
the
topological
group
Π
tp
Ÿ
by
δ
[which
induces
the
identity
on
both
the
quotient
Π
tp
[μ
N
]
Π
tp
and
the
kernel
Ÿ
Ÿ
of
this
quotient].
Then
α̈
δ
extends
to
an
automorphism
α
δ
∈
Aut(Π
tp
Y
[μ
N
])
which
tp
induces
the
identity
on
both
the
quotient
Π
tp
Y
[μ
N
]
Π
Y
and
the
kernel
of
this
to
t
Θ
and
preserves
the
subgroup
D
Y
⊆
quotient;
conjugation
by
α
δ
maps
s
Θ
Ÿ
Ÿ
Out(Π
tp
Y
[μ
N
]).
def
def
tp
Θ
Θ
alg
(iii)
Write
M
=
(Π
tp
Y
[μ
N
],
D
Y
,
s
Ÿ
)
(respectively,
B
=
(Π
Y
[μ
N
],
D
Y
,
s
Ÿ
,
s
Ÿ
))
for
the
model
mono-theta
(respectively,
bi-theta)
environment
constructed
in
the
above
discussion.
Then
every
automorphism
of
M
(respectively,
B)
determines
an
automorphism
of
Π
tp
Y
[cf.
Proposition
2.11,
(ii)],
hence
an
automorphism
of
tp
Π
tp
X
=
Aut(Π
Y
)
×
Out(Π
tp
)
Im(D
Y
)
Y
—
where
“Im(−)”
denotes
the
image
in
Out(Π
tp
Y
)
—
as
well
as
[by
considering
the
cuspidal
decomposition
groups]
an
automorphism
of
the
set
of
cusps
of
Y
.
Relative
to
the
labels
∈
Z
on
these
cusps
[cf.
Corollary
2.9
and
the
discussion
preceding
46
SHINICHI
MOCHIZUKI
it],
this
automorphism
induces
an
automorphism
∈
(l
·
Z)
{±1}
of
Z.
Moreover,
the
resulting
homomorphism
Aut(M)
→
(l
·
Z)
{±1}
(respectively,
Aut(B)
→
(l
·
Z)
{±1})
is
surjective
(respectively,
has
image
Im
N
satisfying
(N
·
l
·
Z)
{±1}
⊆
Im
N
⊆
def
def
(N
†
·
l
·
Z)
{±1}
⊆
Z
{±1}
—
where
N
†
=
N
if
N
is
odd,
N
†
=
N/2
if
N
is
even).
Θ
Proof.
First,
we
consider
assertion
(i).
Observe
that
the
group
(Δ
tp
Y
)
[μ
N
]
is
abelian,
and
that
[since
G
K
is
center-free
—
cf.,
e.g.,
[Mzk2],
Theorem
1.1.1,
(ii)]
×
restrict
to
the
identity
on
the
automorphisms
“γ”
of
Π
tp
Y
[μ
N
]
arising
from
K
Θ
(Δ
tp
Y
)
[μ
N
].
Thus,
one
computes
easily
that
assertion
(i)
follows
immediately
from
Proposition
2.12,
(ii),
in
the
case
where
Δ
∗
=
Δ
tp
X
.
Next,
we
consider
assertion
(ii).
It
is
immediate
from
the
definitions
that
to
t
Θ
.
Since
the
outer
action
of
Gal(Y
/X)
(
∼
conjugation
by
α̈
δ
maps
s
Θ
=
l
·
Z)
Ÿ
Ÿ
alg
on
Π
tp
Y
[μ
N
]
fixes
the
section
s
Ÿ
,
up
to
μ
N
-conjugacy,
it
follows
that
the
difference
cocycle
δ
determines
a
cohomology
class
of
,
μ
N
)
H
1
(Π
tp
Ÿ
that
lies
in
the
submodule
generated
by
the
Kummer
classes
of
K
×
and
“
Ü
2l·(1/l)
=
Ü
2
”
[cf.
Proposition
1.5,
(ii),
(iii)].
Here,
we
note
that
the
factor
of
“1/l”
in
the
exponent
of
Ü
arises
from
the
fact
that
to
work
with
η̈
Θ,l·Z×μ
2
amounts
to
working
with
l-th
roots
of
theta
functions
[cf.
the
discussion
preceding
Definition
2.7];
the
factor
of
“l”
arises
from
the
factor
of
l
in
“l
·
Z”.
Since,
moreover,
the
meromorphic
function
“
Ü
2
”
on
Ÿ
descends
to
Y
,
we
thus
conclude
that
δ
extends
to
a
cocycle
of
Π
tp
Y
with
coefficients
in
μ
N
,
hence
that
α̈
δ
extends
to
an
automorphism
α
δ
∈
Aut(Π
tp
Y
[μ
N
])
which
induces
the
identity
on
both
tp
the
quotient
Π
tp
Y
[μ
N
]
Π
Y
and
the
kernel
of
this
quotient.
Since
the
action
by
an
element
of
Gal(Y
/X)
clearly
maps
Ü
2
to
a
K
×
-multiple
of
Ü
2
,
it
thus
follows
that
conjugation
by
α
δ
preserves
D
Y
⊆
Out(Π
tp
Y
[μ
N
])
[cf.
the
definition
of
D
Y
!],
as
desired.
This
completes
the
proof
of
assertion
(ii).
Finally,
assertion
(iii)
follows
immediately
from
assertion
(ii)
in
the
mono-theta
case
by
considering
[in
the
context
of
assertion
(ii)]
the
action
of
an
arbitrary
element
of
l
·
Z.
In
the
bi-theta
case,
we
observe
that
if,
in
the
situation
of
assertion
(ii),
t
Θ
Ÿ
is
obtained
as
an
N
·
(l
·
Z)-conjugate
of
s
Θ
,
then
the
cocycle
δ
is
a
coboundary;
in
Ÿ
particular,
[in
this
case]
the
automorphism
α
δ
preserves
the
μ
N
-conjugacy
classes
,
t
Θ
,
s
alg
.
This
shows
that
Im
N
⊇
of
subgroups
determined
by
the
images
of
s
Θ
Ÿ
Ÿ
Ÿ
THE
ÉTALE
THETA
FUNCTION
47
(N
·
l
·
Z)
{±1}.
On
the
other
hand,
the
fact
that
Im
N
⊆
(N
†
·
l
·
Z)
{±1}
follows
immediately
by
considering,
in
light
of
the
cohomology
computation
of
Proposition
1.5,
(i),
the
third
displayed
formula
of
Proposition
1.4,
(ii),
applied
to
the
“mod
N
étale
theta
function”,
which
implies
[cf.
the
computation
applied
in
the
proof
of
assertion
(ii)]
that
for
any
a
·
l
∈
Im
N
[where
a
∈
Z],
we
have
2a
≡
0
(mod
N
).
Remark
2.14.1.
Note
that,
in
the
notation
of
Proposition
2.14,
(ii),
although
the
automorphism
α̈
δ
extends
to
an
automorphism
α
δ
of
Π
tp
Y
[μ
N
],
the
automorphism
2
α
δ
fails
to
extend
to
Π
tp
X
[μ
N
]
[i.e.,
since
Ü
fails
to
descend
from
Y
to
X!];
thus,
it
,
t
Θ
:
Π
tp
→
Π
tp
is
essential
to
work
with
homomorphisms
s
Θ
Y
[μ
N
],
as
opposed
to
Ÿ
Ÿ
Ÿ
tp
composites
of
such
homomorphisms
with
the
natural
inclusion
Π
tp
Y
[μ
N
]
→
Π
X
[μ
N
].
Remark
2.14.2.
Note
that
if,
in
the
situation
of
Proposition
2.14,
one
tries
to
replace
Gal(Y
/X)
by
Gal(Y
/C),
then
one
must
contend
with
the
“inversion
automorphism”
[cf.
Proposition
1.5,
(iii)],
which
maps
Ü
→
Ü
−1
.
This
obliges
one
—
if
one
is
to
retain
the
property
that
“conjugation
by
α
δ
preserves
D
Y
”
—
[μ
N
]
that
arise
to
enlarge
“D
Y
”
so
as
to
include
the
outer
automorphisms
of
Π
tp
Ÿ
from
Kummer
classes
of
integral
powers
of
Ü
4
=
(
Ü
2
)
·
(
Ü
−2
)
−1
.
On
the
other
hand,
if
one
enlarges
D
Y
in
this
fashion,
then
one
verifies
easily
[cf.
the
description
of
the
Kummer
class
of
Ü
in
Proposition
1.5,
(ii)]
that
the
subset
considered
in
Proposition
2.14,
(i),
is
no
longer
contained
in
the
image
of
the
tautological
section
of
(l
·
Δ
Θ
)[μ
N
]
(l
·
Δ
Θ
).
Remark
2.14.3.
The
existence
of
“shifting
automorphisms”
as
in
Proposition
2.14,
(ii)
—
cf.
also
the
mono-theta
portion
of
Proposition
2.14,
(iii)
—
may
be
interpreted
as
the
“nonexistence
of
a
mono-theta-theoretic
basepoint”
[cf.
the
discussion
preceding
Corollary
2.9
concerning
“labels”]
relative
to
the
l
·
Z
action
on
Y
—
i.e.,
the
nonexistence
of
a
“distinguished
irreducible
component
of
the
special
fiber
of
Y”
associated
to
the
data
constituted
by
a
mod
N
mono-theta
environment.
On
the
other
hand,
the
description
of
the
poles
of
the
theta
function
[cf.
Proposition
1.4,
(i)]
already
suggests
that
the
data
constituted
by
a
mod
N
bi-theta
environment
[which
includes,
by
considering
the
“difference”
between
the
subgroups
of
Definition
2.13,
(iii),
(c),
(d),
a
choice
of
a
“specific
mod
N
étale
theta
function”]
does
determine,
in
effect,
a
“basepoint
modulo
N
†
”
[cf.
the
bi-theta
portion
of
Proposition
2.14,
(iii)],
i.e.,
a
distinguished
irreducible
component
of
the
special
fiber
of
Y,
up
to
the
action
of
N
†
·
(l
·
Z).
Proposition
2.15.
(Discrete
Rigidity
and
Z-torsors)
Let
us
regard
N
≥1
as
equipped
with
the
order
relation
arising
from
the
monoid
structure
of
N
≥1
.
[That
is
to
say,
for
M,
M
∈
N
≥1
,
“M
≤
M
”
if
and
only
if
M
|M
,
i.e.,
M
divides
48
SHINICHI
MOCHIZUKI
M
.]
Write
T
for
the
category
whose
objects
T
M
,
where
M
∈
N
≥1
,
are
copies
of
Z
[which
we
think
of
as
torsors
over
Z],
and
whose
morphisms
T
M
→
T
M
,
where
M,
M
∈
N
≥1
satisfy
M
|M
,
are
the
composites
of
the
“identity
morphism”
T
M
=
Z
→
Z
=
T
M
with
an
automorphism
of
T
M
arising
from
the
action
of
an
element
∈
M
·
Z
⊆
Z.
Let
E
⊆
N
≥1
be
a
cofinal,
totally
ordered
subset
of
N
≥1
such
that
1
∈
E.
We
shall
refer
to
as
an
E-system
{S
M
;
β
M
,M
}
M,M
∈E
of
T
any
projective
system
β
M
,M
.
.
.
−→
S
M
−→
S
M
−→
.
.
.
of
objects
and
morphisms
of
T
indexed
by
E;
we
shall
refer
to
as
an
isomorphism
of
E-systems
∼
;
β
M
{S
M
;
β
M
,M
}
M,M
∈E
→
{S
M
,M
}
M,M
∈E
∼
any
collection
of
isomorphisms
α
M
:
S
M
→
S
M
[for
M
∈
E]
of
T
such
that
α
M
◦
β
M
,M
=
β
M
,M
◦
α
M
[for
M,
M
∈
E
such
that
M
|M
].
Then:
(i)
(Groups
of
Automorphisms)
If
M
∈
N
≥1
,
then
we
have
a
natural
∼
isomorphism
Aut
T
(T
M
)
→
M
·
Z.
If
M,
M
∈
N
≥1
,
then
any
morphism
φ
:
T
M
→
T
M
of
T
induces
[via
the
condition
of
compatibility
with
φ]
the
injection
∼
∼
Aut
T
(T
M
)
→
M
·
Z
→
M
·
Z
→
Aut
T
(T
M
)
determined
by
the
natural
inclusion
M
·
Z
⊆
M
·
Z.
def
def
(ii)
(Piecewise
Rigid
E-systems)
Let
S
∗
=
{S
M
;
β
M
,M
}
M,M
∈E
,
S
∗
=
;
β
M
{S
M
,M
}
M,M
∈E
be
arbitrary
E-systems
such
that
S
M
=
S
M
=
T
1
for
all
∼
M
∈
E.
Then
there
exists
an
isomorphism
of
E-systems
S
∗
→
S
∗
.
def
(iii)
(Piecewise
Non-rigid
E-systems
I)
For
S
∗
=
{S
M
;
β
M
,M
}
M,M
∈E
def
S
an
E-system
such
that
S
M
=
T
M
for
all
M
∈
E,
let
us
write
j
M
=
β
M,1
(0)
∈
S
T
1
=
Z
⊆
Z.
Then
the
sequence
{j
M
}
M
∈E
converges
in
Z
to
some
element,
which
S
Moreover,
the
resulting
assignment
we
denote
by
j
∞
∈
Z.
S
S
∗
→
j
∞
∈
Z
—
where
S
∗
=
{S
M
;
β
M
,M
}
M,M
∈E
ranges
over
the
E-systems
such
that
S
M
=
T
M
S
for
all
M
∈
E
—
is
surjective,
and
the
image
of
j
∞
in
Z/Z
depends
only
on
the
isomorphism
class
of
S
∗
as
an
E-system.
(iv)
(Piecewise
Non-rigid
E-systems
II)
There
exist
non-isomorphic
;
β
M
E-systems
{S
M
;
β
M
,M
}
M,M
∈E
,
{S
M
,M
}
M,M
∈E
such
that
S
M
=
S
M
=
T
M
for
all
M
∈
E.
Proof.
Assertions
(i)
and
(ii)
are
immediate
from
the
definitions.
Next,
we
consider
S
}
M
∈E
converges
follows
immediately
assertion
(iii).
The
fact
that
the
sequence
{j
M
choose
a
from
the
definitions.
To
verify
the
asserted
surjectivity,
let
j
∞
∈
Z;
THE
ÉTALE
THETA
FUNCTION
49
sequence
{j
M
}
M
∈E
of
elements
of
Z
such
that
j
M
maps
to
the
image
of
j
∞
in
def
(Z/M
Z),
and
j
1
=
0.
Then
for
M,
M
∈
E
such
that
M
|M
,
we
take
S
M
=
T
M
,
def
S
M
=
T
M
,
β
M
,M
to
be
the
composite
of
the
“identity
morphism”
T
M
=
Z
→
Z
=
T
M
with
the
automorphism
of
T
M
determined
by
the
action
of
j
M
−
j
M
∈
S
”
associated
to
the
resulting
E-
M
·
Z
on
T
M
.
Now
it
is
immediate
that
the
“j
∞
def
system
S
∗
=
{S
M
;
β
M
,M
}
M,M
∈E
is
equal
to
j
∞
,
as
desired.
Finally,
the
fact
that
isomorphic
E-systems
yield
the
same
element
∈
Z/Z
is
immediate
from
the
definitions.
This
completes
the
proof
of
assertion
(iii).
Assertion
(iv)
follows
by
taking
S
∗
=
{S
M
;
β
M
,M
}
M,M
∈E
,
S
∗
=
{S
M
;
β
M
,M
}
M,M
∈E
to
be
E-systems
as
S
S
”,
j
∞
”
have
distinct
images
in
Z/Z.
in
assertion
(iii)
such
that
the
associated
“j
∞
Remark
2.15.1.
Proposition
2.15
generalizes
immediately
to
the
case
of
cate-
gories
equivalent
to
the
category
T
.
We
leave
the
routine
details
to
the
reader.
Remark
2.15.2.
Let
T
be
a
“connected
temperoid”
[i.e.,
the
analogue
of
a
Galois
category
for
topological
groups
such
as
tempered
fundamental
groups
—
cf.
[Mzk14],
Definition
3.1,
(ii)].
For
simplicity,
we
suppose
that
T
is
the
temperoid
associated
to
a
topological
group
whose
topology
admits
a
countable
basis
of
open
subgroups.
Then
if
A
→
B
is
a
morphism
of
connected
Galois
objects
of
T
,
then
one
verifies
immediately
that
Aut(A)
acts
transitively
on
Hom
T
(A,
B).
In
particular,
def
def
[cf.
Proposition
2.15,
(ii)]
if
A
=
{A
i
}
i∈N
,
B
=
{B
j
}
j∈N
are
cofinal
[i.e.,
among
the
connected
objects
of
T
]
projective
systems
of
connected
Galois
objects
of
T
indexed
by
N
[equipped
with
its
usual
ordering],
then
there
exists
an
isomorphism
∼
of
projective
systems
A
→
B
[which
does
not
necessarily
induce
an
isomorphism
between
the
various
A
i
,
B
j
].
It
is
this
sort
of
projective
system
that
is
implicitly
used
in
the
proof
of
[Mzk14],
Proposition
3.2,
to
pass
from
the
temperoid
to
its
associated
fundamental
group.
Corollary
2.16.
(Profinite
Non-discrete-ness
of
Bi-theta
Environments)
Fix
some
member
η̈
Θ
of
the
collection
of
[cocycles
determined
by
the
collection
of
]
classes
η̈
Θ,l·Z×μ
2
[cf.
the
discussion
preceding
Proposition
2.14]
in
H
1
(Π
tp
,
l
·
Δ
Θ
).
For
M
∈
N
≥1
,
write
Ÿ
B
M
for
the
model
bi-theta
environment
that
arises
from
[cf.
Definition
2.13,
(iii)]
the
reduction
of
this
η̈
Θ
modulo
M
;
Π
M
[μ
M
]
Π
M
for
the
portion
of
the
data
B
M
constituted
by
the
topological
group
[together
with
its
natural
surjection]
—
cf.
Definition
2.13,
(iii),
(a)
[so
Π
M
may
be
thought
of
as
a
copy
of
Π
tp
Y
];
Π̈
M
⊆
Π
M
for
the
subgroup
which
is
the
image
in
Π
M
of
the
theta
section
—
cf.
Definition
50
SHINICHI
MOCHIZUKI
2.13,
(iii),
(c)
[so
Π̈
M
may
be
thought
of
as
a
copy
of
Π
tp
].
Let
E
⊆
N
≥1
be
a
Ÿ
cofinal,
totally
ordered
subset
of
N
≥1
[cf.
Proposition
2.15]
such
that
1
∈
E.
Thus,
we
obtain
a
natural
projective
system
of
bi-theta
environments
β
M
,M
.
.
.
−→
B
M
−→
B
M
−→
.
.
.
Then
there
exists
a
projective
—
where
M,
M
∈
E;
M
|M
.
Let
j
∞
∈
l
·
Z.
system
of
bi-theta
environments
γ
M
,M
.
.
.
−→
B
M
−→
B
M
−→
.
.
.
—
where
M,
M
∈
E;
M
|M
—
such
that
the
following
properties
hold:
(a)
for
each
γ
M
,M
,
there
exist
automorphisms
α,
α
of
the
bi-theta
environments
B
M
,
B
M
,
respectively,
[where
α,
α
may
depend
on
the
pair
(M,
M
)]
such
that
γ
M
,M
=
α
◦
β
M
,M
◦
α
;
(b)
the
classes
[indexed
by
M
]
of
,
l
·
Δ
Θ
)
H
1
(Π̈
1
,
l
·
Δ
Θ
)
=
H
1
(Π
tp
Ÿ
obtained
by
transporting
the
difference
of
the
algebraic
and
theta
sections
of
B
M
∼
down
to
Π̈
1
via
the
isomorphism
Π̈
M
→
Π̈
1
induced
by
γ
M,1
converge
to
the
element
∧
of
H
1
(Π
Ÿ
∧
,
l
·
Δ
Θ
)
[where
Ÿ
is
the
profinite
étale
covering
determined
by
Ÿ
∧
—
cf.
Remark
1.6.4]
given
by
the
j
∞
-conjugate
of
the
restriction
of
η̈
Θ
to
Ÿ
.
Proof.
In
light
of
the
symmetries
of
bi-theta
environments
[cf.
Proposition
2.14,
(iii)],
Corollary
2.16
follows
immediately
from
Proposition
2.15,
(iii).
Remark
2.16.1.
(i)
Observe
that
the
property
discussed
in
Corollary
2.16
[i.e.,
Proposition
2.15,
(iii)]
is
indicative
of
a
fundamental
qualitative
difference
between
mono-
and
bi-theta
environments.
Indeed,
if
one
allows
the
integer
N
≥
1
to
vary
[multiplicatively,
i.e.,
in
N
≥1
],
then
the
various
resulting
mono-
and
bi-theta
environments
naturally
determine
projective
systems.
Moreover,
it
is
natural
to
think
of
each
of
the
mod
N
mono-
or
bi-theta
environments
appearing
in
these
projective
systems
as
only
being
known
up
to
isomorphism
[cf.
Remarks
5.12.1,
5.12.2
in
§5
below
for
more
on
this
point].
From
this
point
of
view,
Proposition
2.15,
(i),
(ii),
when
applied
to
mono-theta
environments
[cf.
Corollary
2.19,
(ii),
(iii),
below],
asserts,
in
effect,
that:
If
one
works
with
this
projective
system
of
mono-theta
environments,
then
in
light
of
the
compatibility
of
the
various
[collections
of
subgroups
deter-
mined
by
the
image
of
the]
theta
sections
of
the
mono-theta
environments
in
the
projective
system,
the
various
mod
N
étale
theta
classes
determine,
in
the
projective
limit,
a
single
“discrete”
l
·
Z-torsor
THE
ÉTALE
THETA
FUNCTION
51
whose
reduction
modulo
N
[i.e.,
the
result
of
applying
a
change
of
struc-
ture
group
via
the
homomorphism
l
·
Z
l
·
Z/N
·
l
·
Z]
appears
in
the
mod
N
mono-theta
environment.
By
contrast,
Corollary
2.16
implies
that
if
one
tries
to
carry
out
such
a
construction
in
the
case
of
bi-theta
environments,
then
since
the
projective
system
in
question
gives
rise
to
a
“basepoint
indeterminacy”
[cf.
Proposition
2.14,
(iii)],
for
the
mod
N
bi-theta
environment
of
the
system,
given
by
some
group
lying
between
the
groups
N
·
l
·
Z,
N
†
·
l
·
Z,
the
resulting
projective
limit
necessarily
leads
to
a
“torsor
of
def
=
l
·
Z
⊗
Z.
possible
basepoints”
over
the
“non-discrete”
profinite
limit
group
l
·
Z
Put
another
way,
the
crucial
“shifting
symmetry”
that
exists
in
the
case
of
a
mono-
theta
environment
[cf.
Proposition
2.14,
(ii);
Remark
2.14.3]
gives
rise
to
a
“constant
[i.e.,
independent
of
N
]
l
·
Z-indeterminacy”,
hence
implies
precisely
that,
in
the
mono-theta
case,
the
problem
of
“finding
a
common
basepoint”
for
the
various
(l
·
Z/N
·
l
·
Z)-torsors
that
appear
in
the
projective
system
amounts
to
the
issue
of
trivializing
a
torsor
over
the
projective
limit
∼
lim
←−
(l
·
Z/l
·
Z)
=
{0}
N
—
which
remains
“discrete”
—
whereas
in
the
case
of
a
bi-theta
environment,
the
corresponding
torsor
is
a
torsor
over
the
projective
limit
∼
lim
←−
(l
·
Z/N
·
l
·
Z)
=
l
·
Z
N
—
which
is
“essentially
profinite”,
hence,
in
particular,
“non-discrete”.
(ii)
Note
that
the
“non-discreteness”
discussed
in
(i),
which
arose
from
the
lack
of
symmetry
of
a
bi-theta
environment,
by
comparison
to
a
mono-theta
environment
[cf.
Proposition
2.14,
(iii)],
cannot
be
remedied
by,
for
instance,
considering
“multi-
bi-theta
environments”
in
which
instead
of
considering
just
one
theta
section,
one
considers
an
entire
(l
·
Z/N
·
l
·
Z)-orbit
of
theta
sections.
Indeed,
if
one
considers
such
orbits,
then
the
resulting
projective
system
requires
one
to
consider
[not
an
l
·Z-orbit
of
an
étale
theta
function,
but
rather]
an
entire
orbit
over
the
non-discrete
profinite
group
∼
lim
←−
(l
·
Z/N
·
l
·
Z)
=
l
·
Z
N
of
étale
theta
functions
—
i.e.,
one
must
contend
with
essentially
the
same
“non-
discreteness”
phenomenon
as
was
discussed
in
(i).
(iii)
The
“non-discreteness”
phenomenon
discussed
in
(i)
may
also
be
formu-
lated
from
a
more
“cohomological”
point
of
view,
by
considering
the
first
derived
lim
functor
R
1
lim
←−
of
the
projective
limit
←
−
.
That
is
to
say,
if
one
considers
the
exact
sequence
of
projective
systems
of
modules
{0
→
N
·
l
·
Z
→
l
·
Z
→
(l
·
Z/N
·
l
·
Z)
→
0}
N≥1
obtained
by
allowing
the
integer
N
≥
1
to
vary
multiplicatively
[and
taking
the
transition
morphisms
to
be
the
morphisms
determined
by
identifying
the
various
52
SHINICHI
MOCHIZUKI
copies
of
l
·
Z],
then
the
[unique
nontrivial]
connecting
homomorphism
of
the
long
exact
sequence
associated
to
the
derived
functors
of
“lim
←−
”
yields
a
natural
isomor-
phism
∼
·
Z)
→
R
1
lim
(l
·
Z)/(l
←−
{N
·
l
·
Z}
N≥1
of
the
associated
“R
1
lim
←−
”
with
the
“nonarchimedean
solenoid”
(l
·
Z)/(l
·
Z).
That
·
Z)
is
essentially
equivalent
to
the
is
to
say,
the
nonvanishing
of
this
group
(l
·
Z)/(l
“non-discreteness”
phenomenon
discussed
in
(i).
Remark
2.16.2.
Although
the
present
paper
is
essentially
only
concerned
with
the
“local
theory”
of
the
theta
function
[i.e.,
over
finite
extensions
of
Q
p
],
frequently
in
applications
[cf.
[Mzk4],
[Mzk5]]
it
is
of
interest
to
develop
the
local
theory
in
such
a
way
that
it
may
be
related
naturally
to
the
“global
theory”
[i.e.,
over
number
fields].
In
such
situations,
one
is
typically
obligated
to
contend
with
some
sort
of
homomorphism
of
topological
groups
φ
:
Π
tp
X
→
Π
X
F
relating
the
tempered
fundamental
group
of
the
smooth
log
curve
X
log
[appearing
in
the
theory
of
the
present
paper]
to
the
profinite
fundamental
group
of
a
smooth
log
curve
X
log
over
a
number
field
F
such
that
X
log
is
obtained
from
X
log
by
F
F
base-changing
to
some
completion
K
=
F
v
of
F
at
a
finite
prime
v.
Moreover,
typically,
one
must
assume
that
φ
is
only
given
up
to
composition
with
an
inner
automorphism
[i.e.,
as
an
“outer
homomorphism”].
Alternatively,
one
may
think
of
φ
“category-theoretically”
via
its
associated
morphism
of
temperoids
[cf.
[Mzk14],
Proposition
3.2]
T
→
T
F
—
i.e.,
a
functor
T
F
→
T
[obtained
by
associating
to
a
Π
X
-set
the
Π
tp
X
-set
deter-
F
mined
by
composing
with
φ],
which
is
typically
only
determined
up
to
isomorphism.
In
this
situation,
connected
tempered
coverings
of
X
log
[e.g.,
a
finite
étale
covering
of
Y
log
],
which
correspond
to
open
subgroups
H
⊆
Π
tp
X
,
are
subject
to
an
indeter-
minacy
with
respect
to
conjugation
by
elements
of
the
normalizer
N
Π
X
(Im(H))
F
of
the
image
of
H
in
Π
X
—
i.e.,
as
opposed
to
just
the
“weaker”
indeterminacy
F
with
respect
to
conjugation
by
elements
of
the
normalizer
N
Π
tp
(H),
which
arises
X
from
working
with
the
topological
group
Π
tp
X
up
to
inner
automorphism.
In
this
situation,
since
the
two
normalizers
in
question
in
fact
coincide
—
i.e.,
we
have
N
Π
tp
(H)
=
N
Π
X
(Im(H))
X
F
[by
Lemma
2.17,
(ii),
below;
the
well-known
fact
that
the
absolute
Galois
group
G
F
v
is
equal
to
its
own
normalizer
in
the
absolute
Galois
group
G
F
—
cf.,
e.g.,
[Mzk2],
THE
ÉTALE
THETA
FUNCTION
53
Theorem
1.1.1,
(i)]
—
in
the
present
situation,
this
state
of
affairs
does
not
in
fact
result
in
any
further
indeterminacy
[by
comparison
to
the
strictly
local
situation].
Moreover,
the
above
equality
of
normalizers
also
shows
that
replacing
Π
tp
X
by
H,
for
instance,
does
not
result
[in
the
present
local/global
situation]
in
any
reduction
in
the
indeterminacy
to
which
H
is
subject.
Thus,
when
H
corresponds
to
a
finite
étale
covering
of
Y
log
,
the
corresponding
covering
will
always
be
subject
to
an
indeterminacy
with
respect
to
the
action
of
some
finite
index
subgroup
of
Z
[an
indeterminacy
which
results
in
the
sort
of
situation
discussed
in
Remark
2.16.1].
Thus,
in
summary:
The
indeterminacy
which
results
in
the
phenomena
discussed
in
Remark
2.16.1
may
be
regarded
as
the
inevitable
result
of
attempting
to
accommo-
date
simultaneously
the
“two
mutually
alien
copies
of
Z”
constituted
by
the
geometric
Galois
group
Z
and
the
arithmetic
global
base
Z
⊆
Q
⊆
F
.
[Here,
we
remark
that
the
“mutual
alienness”
of
these
two
copies
of
Z
arises
from
the
fact
that
[non-finite]
tempered
coverings
only
exist
p-adically,
hence
fail
to
descend
to
coverings
defined
over
a
number
field.]
Remark
2.16.3.
Relative
to
the
analogy
between
Galois
group
actions
and
differentials
[cf.
the
discussion
of
[Mzk4]],
the
equality
of
the
normalizers
discussed
in
Remark
2.16.2
may
be
thought
of
as
a
sort
of
group-theoretic
version
of
the
condition
that
the
map
from
a
finite
prime
of
a
number
field
to
the
global
number
field
be
“unramified”.
Lemma
2.17.
(Discrete
Normalizers)
If
G
1
is
a
subgroup
of
a
group
G
2
,
then
write
N
G
2
(G
1
)
for
the
normalizer
of
G
1
in
G
2
.
Then:
(i)
Let
F
be
a
group
that
contains
a
normal
subgroup
of
finite
index
G
⊆
F
such
that
G
is
a
free
discrete
group
of
finite
rank,
H
⊆
F
a
subgroup
such
for
the
profinite
completions
that
the
group
H
G
is
nonabelian.
Write
F
,
G
of
F
,
G
[so
we
have
a
natural
inclusion
F
→
F
].
Then
N
F
(H)
=
N
F
(H).
(ii)
Let
Π
be
the
tempered
fundamental
group
of
a
hyperbolic
orbicurve
over
a
for
the
profinite
finite
extension
K
of
Q
p
,
H
⊆
Π
an
open
subgroup.
Write
Π
completion
of
Π.
Then
N
Π
(H)
=
N
Π
(H).
Proof.
The
proof
of
assertion
(i)
is
similar
to
[but
slightly
more
involved
than]
the
proof
of
the
case
H
=
G
=
F
discussed
in
[André],
Lemma
3.2.1:
By
replacing
H
we
may
assume
that
H
⊆
G.
Let
{x
i
}
i∈I
[where
I
is
some
by
H
G
=
H
G,
index
set
of
cardinality
≥
2]
be
a
set
of
generators
of
H,
a
∈
N
F
(H).
Now
let
us
fix
two
distinct
elements
i
1
,
i
2
∈
I
[so
x
i
1
,
x
i
2
generate
a
free
subgroup
of
G
of
rank
2].
Then
there
exists
a
subgroup
J
⊆
G
⊆
F
of
finite
index
such
that
x
i
1
,
x
i
2
∈
J,
and,
moreover,
x
i
1
,
x
i
2
appear
in
some
collection
of
free
generators
of
J
[cf.
[Mzk14],
Corollary
1.6,
(ii)].
In
particular,
for
each
j
=
1,
2,
the
centralizer
54
SHINICHI
MOCHIZUKI
of
x
i
j
in
the
profinite
completion
J
of
J
is
topologically
generated
by
x
i
j
[cf.,
e.g.,
[Mzk15],
Proposition
1.2,
(ii)].
Moreover,
since
J
is
of
finite
index
in
F
,
it
follows
that
there
exists
a
b
∈
F
such
that
a
∈
b
·
J
(⊆
F
).
In
particular,
it
follows
that,
Now
for
each
j
=
1,
2,
b
−1
ax
i
j
a
−1
b
∈
F
J
=
J
is
conjugate
to
x
i
j
∈
J
in
J.
by
a
classical
result
of
P.
Stebe
[cf.
[LynSch],
Proposition
4.9],
it
follows
that
J
is
“conjugacy-separated”,
hence
that
for
each
j
=
1,
2,
there
exists
an
a
j
∈
b
·
J
(⊆
F
)
−1
a
j
belongs
to
such
that
ax
i
j
a
−1
=
a
j
x
i
j
a
−1
j
.
Thus,
for
each
j
=
1,
2,
c
j
=
a
λ
j
hence
is
of
the
form
x
,
for
some
λ
j
∈
Z.
But
this
the
centralizer
of
x
i
j
in
J,
i
j
−1
implies
that
c
−1
J
=
J,
hence
[for
instance,
by
considering
the
2
c
1
=
a
2
a
1
∈
F
that
c
1
,
c
2
∈
J
⊆
F
,
so
a
∈
F
,
as
desired.
image
of
c
1
,
c
2
in
the
abelianization
of
J]
Assertion
(ii)
now
follows
immediately
from
assertion
(i)
by
applying
assertion
(i)
to
quotients
of
Π
by
characteristic
open
subgroups
of
Π,
which
contain
finite
rank
free
normal
subgroups
of
finite
index.
def
We
are
now
ready
to
state
the
two
main
results
of
the
present
§2
concerning
mono-theta
environments.
Corollary
2.18.
(Group-theoretic
Construction
of
Mono-theta
Environ-
ments)
Let
N
≥
1
be
an
integer;
X
log
a
smooth
log
curve
of
type
(1,
(Z/lZ)
Θ
)
over
a
finite
extension
K
of
Q
p
,
where
l
and
p
are
odd,
such
that
K
=
K̈;
η̈
Θ,l·Z×μ
2
an
associated
orbit
of
l-th
roots
of
étale
theta
functions;
Y
log
→
X
log
,
log
→
Y
log
the
corresponding
coverings
[as
in
the
above
discussion];
(l
·
Δ
Θ
),
Ÿ
tp
Θ
tp
Θ
tp
Θ
tp
Θ
(Δ
tp
X
)
,
(Π
X
)
,
(Δ
Y
)
,
(Π
Y
)
the
resulting
subquotients
of
Π
X
[as
in
the
above
discussion];
def
Δ
[μ
N
]
=
Ker((l
·
Δ
Θ
)[μ
N
]
→
(l
·
Δ
Θ
))
[i.e.,
the
“μ
N
”
of
“[μ
N
]”];
η̈
Θ,l·Z×μ
2
[μ
N
]
the
collection
of
classes
of
H
1
(Π
tp
,
Δ
[μ
N
]
)
obtained
by
applying
the
natural
surjec-
Ÿ
tion
(l
·
Δ
Θ
)
Δ
[μ
N
]
to
η̈
Θ,l·Z×μ
2
;
D
Y
⊆
Out(Π
tp
Y
[μ
N
])
×
∼
the
subgroup
of
Out(Π
tp
Y
[μ
N
])
generated
by
the
image
of
K
,
Gal(Y
/X)
(
=
l
·
Z)
[cf.
Definition
2.13,
(i)];
:
Π
tp
→
Π
tp
s
alg
Y
[μ
N
];
Ÿ
Ÿ
s
Θ
:
Π
tp
→
Π
tp
Y
[μ
N
]
Ÿ
Ÿ
the
resulting
mod
N
algebraic
and
theta
sections
[determined
by
a
cocycle
repre-
senting
a
member
of
the
collection
of
classes
η̈
Θ,l·Z×μ
2
];
def
Θ
M
N
=
(Π
tp
Y
[μ
N
],
D
Y
,
s
Ÿ
)
THE
ÉTALE
THETA
FUNCTION
55
the
resulting
mod
N
model
mono-theta
environment
[which,
by
Proposition
2.14,
(ii),
is
independent,
up
to
isomorphism
over
the
identity
of
Π
tp
Y
,
of
the
choice
of
[a
cocycle
representing
a
member
of
the
collection
of
classes]
η̈
Θ,l·Z×μ
2
,
among
its
multiples
by
a
2l-th
root
of
unity].
Then:
(i)
(Theta-related
Subquotients)
Let
Π
•
X
be
a
topological
group
that
is
isomorphic
to
Π
tp
X
.
Then
there
exists
a
“functorial
group-theoretic
algorithm”
—
i.e.,
an
algorithm
that
invokes
only
the
structure
of
Π
•
X
as
an
abstract
topolog-
ical
group,
is
functorial
with
respect
to
isomorphisms
of
topological
groups,
and
is
devoid
of
any
reference
to
any
isomorphisms
of
Π
•
X
with
Π
tp
X
—
for
constructing
subquotients
Π
•
Y
;
Π
•
Ÿ
;
(Π
•
X
)
G
•
K
;
(l
·
Δ
•
Θ
);
(Δ
•
X
)
Θ
;
(Π
•
X
)
Θ
;
(Δ
•
Y
)
Θ
;
(Π
•
Y
)
Θ
of
Π
•
X
,
as
well
as
a
collection
of
subgroups
of
Π
•
X
for
each
element
of
(Z/lZ)
±
,
∼
which
have
the
property
that
any
isomorphism
Π
•
X
→
Π
tp
X
maps
the
above
subquo-
tients,
respectively,
to
the
subquotients
Π
tp
Y
;
Π
tp
;
Ÿ
(Π
tp
X
)
G
K
;
(l
·
Δ
Θ
);
Θ
(Δ
tp
X
)
;
Θ
(Π
tp
X
)
;
Θ
(Δ
tp
Y
)
;
Θ
(Π
tp
Y
)
of
Π
tp
X
,
and
the
above
collection
of
subgroups
to
the
collection
of
cuspidal
de-
composition
groups
of
Π
tp
X
determined
by
the
label
∈
(Z/lZ)
±
[cf.
Corollary
2.9].
(ii)
(From
Topological
Groups
to
Mono-theta
Environments)
In
the
situation
of
(i),
there
exists
a
“functorial
group-theoretic
algorithm”
for
con-
def
structing
a
mod
N
mono-theta
environment
M
•
=
(Π
•
,
D
Π
•
,
s
Θ
Π
•
),
where
Π
•
=
Π
•
Y
×
G
•
K
{(l
·
Δ
Θ
)
⊗
(Z/N
Z)}
G
•
K
def
[cf.
Definition
2.10],
“up
to
isomorphism”.
More
precisely,
there
exists
a
“functorial
group-theoretic
algorithm”
for
constructing
a
collection
of
mod
N
mono-theta
environments
{M
•
ι
}
ι∈I
,
where
M
•
ι
=
(Π
•
,
D
Π
•
,
(s
Θ
Π
•
)
ι
),
such
that,
for
•
∼
•
ι
1
,
ι
2
∈
I,
there
exists
an
isomorphism
M
ι
1
→
M
ι
2
that
induces
the
identity
on
the
quotient
Π
•
Π
•
Y
.
(iii)
(From
Mono-theta
Environments
to
Topological
Groups)
Let
•
def
M
=
(Π
•
,
D
Π
•
,
s
Θ
Π
•
)
be
a
mod
N
mono-theta
environment
isomorphic
to
M
N
.
Then
there
exists
a
“functorial
group-theoretic
algorithm”
—
i.e.,
an
algo-
rithm
that
invokes
only
the
structure
of
M
•
as
an
abstract
mono-theta
environment,
is
functorial
with
respect
to
isomorphisms
of
mono-theta
environments,
and
is
de-
void
of
any
reference
to
any
isomorphisms
of
M
•
with
M
N
—
for
constructing
a
quotient
Π
•
Y
56
SHINICHI
MOCHIZUKI
∼
of
Π
•
which
has
the
property
that
any
isomorphism
M
•
→
M
N
maps
this
quotient,
respectively,
to
the
quotient
Π
tp
Y
∼
•
of
Π
tp
Y
[μ
N
].
Moreover,
any
such
isomorphism
M
→
M
N
also
induces
an
isomor-
phism
of
def
Π
•
X
=
Aut(Π
•
Y
)
×
Out(Π
•
Y
)
Im(D
Π
•
)
—
where
“Im(−)”
denotes
the
image
in
Out(Π
•
Y
)
[cf.
Proposition
2.11,
(ii)];
the
topology
of
Π
•
X
is
the
topology
determined
by
taking
∼
Π
•
Y
→
Aut(Π
•
Y
)
×
Out(Π
•
Y
)
{1}
⊆
Π
•
X
•
to
be
an
open
subgroup
—
with
Π
tp
X
.
Finally,
M
is
isomorphic
to
the
mono-theta
environment
obtained
by
applying
the
algorithm
of
(ii)
to
Π
•
X
,
via
an
isomorphism
that
induces
the
identity
on
Π
•
Y
.
def
)
be
(iv)
(Lifting
Isomorphisms)
For
=
α,
β,
let
M
=
(Π
,
D
Π
,
s
Θ
Π
•
a
mod
N
mono-theta
environment;
Π
X
be
the
topological
group
“Π
X
”
of
(iii)
[i.e.,
by
taking
“M
•
”
to
be
M
].
Then
the
natural
map
[cf.
(iii)]
β
Isom
μ
(M
α
,
M
β
)
→
Isom(Π
α
X
,
Π
X
)
—
where
the
superscripted
“μ”
denotes
the
set
of
μ
N
-conjugacy
classes
of
iso-
morphisms
—
is
surjective
with
fibers
of
cardinality
1
(respectively,
2)
if
N
is
odd
(respectively,
even).
In
particular,
for
any
positive
integer
M
such
that
M
|N
,
the
induces
a
natural
ho-
mod
M
mono-theta
environment
M
M
determined
by
M
μ
μ
momorphism
Aut
(M
)
→
Aut
(M
M
)
with
normal
image,
whose
kernel
and
cokernel
have
the
same
cardinalities
[≤
2],
respectively,
as
the
kernel
and
coker-
nel
of
the
homomorphism
Hom(Z/2Z,
Z/N
Z)
→
Hom(Z/2Z,
Z/M
Z)
induced
by
the
natural
surjection
Z/N
Z
Z/M
Z
[hence
is
a
bijection
if
N/M
is
odd].
Proof.
First,
we
consider
assertion
(i).
An
algorithm
for
constructing
the
subquo-
tients
Π
•
Y
;
Π
•
Ÿ
;
(l
·
Δ
•
Θ
);
(Δ
•
X
)
Θ
;
(Π
•
X
)
Θ
;
(Δ
•
Y
)
Θ
;
(Π
•
Y
)
Θ
(respectively,
(Π
•
X
)
G
•
K
)
is
described
in
the
proofs
of
Propositions
1.8,
2.4
[cf.
also
the
definitions
of
the
various
coverings
involved!]
(respectively,
in
the
proof
of
[Mzk2],
Lemma
1.3.8).
An
algorithm
for
constructing
the
labels
of
cuspidal
decomposition
groups
is
described
in
the
proof
of
Corollary
2.9
[cf.
also
the
proof
of
[Mzk2],
Lemma
2.3].
This
completes
the
proof
of
assertion
(i).
Assertion
(ii)
follows
immediately
from
the
construction
of
the
model
mono-theta
environment
in
the
discussion
preceding
Definition
2.13;
the
fact
that
the
various
choices
involved
in
this
construction
yield
isomorphic
mono-theta
environments
via
isomorphisms
that
THE
ÉTALE
THETA
FUNCTION
57
induce
the
identity
on
the
quotient
Π
•
Π
•
Y
is
precisely
the
content
of
Proposition
2.14,
(ii).
Next,
we
consider
assertion
(iii).
The
algorithm
for
constructing
the
quotient
Π
Π
•
Y
is
precisely
the
content
of
Proposition
2.11,
(ii);
the
construction
of
Π
•
X
then
follows
immediately,
in
light
of
the
temp-slimness
of
Π
•
X
[cf.
the
proof
of
Proposition
2.11].
The
final
portion
of
assertion
(iii)
[concerning
the
compati-
bility
with
the
algorithm
of
assertion
(ii)]
follows
immediately
from
the
definition
of
a
mono-theta
environment
[cf.
Definition
2.13,
(ii)]
as
“data
isomorphic
to
a
model
mono-theta
environment”
[together
with
the
description
given
in
the
proof
of
assertion
(ii)
of
the
algorithm
of
assertion
(ii)].
•
Finally,
we
consider
assertion
(iv).
First,
we
observe
that
the
functoriality
of
the
“functorial
group-theoretic
algorithm”
of
assertion
(iii)
yields
a
natural
map
β
Isom
μ
(M
α
,
M
β
)
→
Isom(Π
α
X
,
Π
X
).
The
surjectivity
of
this
map
follows
by
applying
the
“functorial
group-theoretic
algorithm”
of
assertion
(ii),
in
light
of
the
final
portion
of
assertion
(iii)
concerning
the
relation
with
the
algorithm
of
assertion
(ii)
[cf.,
especially,
the
fact
that
the
isomorphism
of
mono-theta
environments
appearing
in
this
final
portion
induces
the
identity
on
“Π
•
Y
”].
The
fibers
of
this
map
are
torsors
over
[the
isomorphic
groups]
Ker(Aut
μ
(M
)
→
Aut(Π
X
))
[where
∈
{α,
β}].
To
def
simplify
notation,
let
us
set
M
•
=
M
.
Next,
let
us
observe
that
by
Corollary
2.19,
(i),
below
[where
one
checks
immediately
that
there
are
no
“vicious
circles”
in
the
reasoning],
the
natural
isomorphism
∼
(l
·
Δ
•
Θ
)
⊗
Z/N
Z
→
Ker(Π
•
Π
•
Y
)
is
preserved
by
automorphisms
of
M
•
.
Thus,
Ker(Aut
μ
(M
•
)
→
Aut(Π
•
X
))
—
which
consists
of
automorphisms
that
act
as
the
identity
on
Π
•
Y
,
hence
[by
applying
the
above
natural
isomorphism]
also
on
Ker(Π
•
Π
•
Y
)
—
is
naturally
isomorphic
to
the
group
Hom(Π
•
Y
/Π
•
Ÿ
,
Ker(Π
•
Π
•
Y
))
—
which
is
of
cardinality
1
(respectively,
2)
if
N
is
odd
(respectively,
even).
More-
over,
it
follows
immediately
from
this
description
of
Ker(Aut
μ
(M
•
)
→
Aut(Π
•
X
))
that
the
natural
homomorphism
Aut
μ
(M
•
)
Aut
μ
(M
•
M
)
is
as
described
in
the
statement
of
assertion
(iv).
This
completes
the
proof
of
assertion
(iv).
Remark
2.18.1.
It
follows
immediately
from
Proposition
2.14,
(iii),
that,
for
instance,
the
bijectivity
[i.e.,
“if
N/M
is
odd”]
of
the
latter
portion
of
Corollary
2.18,
(iv),
is
false
for
bi-theta
environments.
Remark
2.18.2.
Thus,
in
a
word,
Corollary
2.18
may
be
interpreted
as
asserting
that
a
mono-theta
environment
may
be
regarded
as
an
object
naturally
constructed
from/associated
to
the
tempered
fundamental
group.
On
the
other
hand,
as
we
shall
58
SHINICHI
MOCHIZUKI
see
in
§5,
a
mono-theta
environment
also
appears
as
an
object
this
may
be
naturally
constructed
from/associated
to
a
certain
Frobenioid.
In
fact:
One
of
the
main
motivating
reasons,
from
the
point
of
view
of
the
author,
for
the
introduction
of
the
notion
of
a
mono-theta
environment
was
pre-
cisely
the
fact
that
it
provides
a
convenient
common
ground
for
relating
the
[tempered-]étale-theoretic
and
Frobenioid-theoretic
approaches
to
the
theta
function.
This
point
of
view
will
be
discussed
in
more
detail
in
Remark
5.10.1
in
§5
below.
Corollary
2.19.
(Rigidity
Properties
of
Mono-theta
Environments)
In
the
notation
of
Corollary
2.18:
def
(i)
(Cyclotomic
Rigidity)
Let
M
•
=
(Π
•
,
D
Π
•
,
s
Θ
Π
•
)
be
a
mod
N
mono-
theta
environment
isomorphic
to
M
N
.
Thus,
by
Corollary
2.18,
(iii),
we
obtain
a
topological
group
Π
•
X
from
M
•
to
which
Corollary
2.18,
(i),
(ii),
may
be
ap-
plied.
Then
there
exists
a
“functorial
group-theoretic
algorithm”
—
i.e.,
an
algorithm
that
invokes
only
the
structure
of
M
•
as
an
abstract
mono-theta
environ-
ment,
is
functorial
with
respect
to
isomorphisms
of
mono-theta
environments,
and
is
devoid
of
any
reference
to
any
isomorphisms
of
M
•
with
M
N
—
for
constructing
subquotients
Π
•
|
(l·Δ
•
Θ
)
⊆
Π
•
|
(Δ
•
Y
)
Θ
⊆
Π
•
|
(Π
•
Y
)
Θ
[cf.
the
notation
of
Corollary
2.18,
(i)]
of
Π
•
which
have
the
property
that
any
∼
isomorphism
M
•
→
M
N
maps
these
subquotients,
respectively,
to
the
subquotients
tp
Θ
Θ
(l
·
Δ
Θ
)[μ
N
]
⊆
(Δ
tp
Y
)
[μ
N
]
⊆
(Π
Y
)
[μ
N
]
of
Π
tp
Y
[μ
N
].
Moreover,
there
exists
a
“functorial
group-theoretic
algorithm”
for
constructing
two
splittings
of
the
natural
surjection
Π
•
|
(l·Δ
•
Θ
)
(l
·
Δ
•
Θ
)
—
hence,
in
particular,
[by
forming
the
difference
of
these
two
splittings]
an
iso-
morphism
of
cyclotomes
∼
((l
·
Δ
•
Θ
)
)
(l
·
Δ
•
Θ
)
⊗
(Z/N
Z)
→
Π
•
μ
=
Ker(Π
•
|
(l·Δ
•
Θ
)
(l
·
Δ
•
Θ
))
def
∼
—
which
have
the
property
that
any
isomorphism
M
•
→
M
N
maps
these
two
split-
tings,
respectively,
to
the
two
splittings
of
the
surjection
(l
·
Δ
Θ
)[μ
N
]
(l
·
Δ
Θ
)
,
s
Θ
[and
hence
the
above
determined
by
the
algebraic
and
theta
sections
s
alg
Ÿ
Ÿ
isomorphism
of
cyclotomes
to
the
natural
isomorphism
of
cyclotomes
determined
,
s
Θ
—
cf.
the
construction
preceding
Definition
2.13].
by
s
alg
Ÿ
Ÿ
THE
ÉTALE
THETA
FUNCTION
59
(ii)
(Discrete
Rigidity)
Let
E
⊆
N
≥1
be
a
cofinal,
totally
ordered
subset
of
N
≥1
[cf.
Proposition
2.15]
such
that
1
∈
E.
Thus,
by
letting
the
integer
N
vary
in
E,
we
obtain
a
natural
projective
system
∗
β
M
,M
.
.
.
−→M
M
−→
M
M
−→
.
.
.
of
model
mono-theta
environments
indexed
by
E
[cf.
Corollary
2.16].
Then
any
projective
system
∗
γ
M
,M
.
.
.
−→M
•
M
−→
M
•
M
−→
.
.
.
—
where
M,
M
∈
E;
M
•
M
is
a
mod
M
mono-theta
environment
—
is
isomorphic
to
the
above
natural
projective
system,
i.e.,
there
exists
a
collection
of
isomorphisms
∼
∗
∗
α
M
:
M
M
→
M
M
such
that
γ
M
,M
◦
α
M
=
α
M
◦
β
M
,M
,
for
M,
M
∈
E
satisfying
M
|M
[cf.
Proposition
2.15,
(ii)].
(iii)
(Constant
Multiple
Rigidity)
Suppose
that
η̈
Θ,l·Z×μ
2
is
of
standard
type
[cf.
Definitions
1.9,
(ii);
2.7].
Let
Π
•
X
be
a
topological
group
that
is
iso-
morphic
to
Π
tp
X
.
Then
there
exists
a
“functorial
group-theoretic
algorithm”
—
i.e.,
an
algorithm
that
invokes
only
the
structure
of
Π
•
X
as
an
abstract
topolog-
ical
group,
is
functorial
with
respect
to
isomorphisms
of
topological
groups,
and
is
devoid
of
any
reference
to
any
isomorphisms
of
Π
•
X
with
Π
tp
X
—
for
constructing
a
collection
of
classes
of
H
1
(Π
•
Ÿ
,
(l
·
Δ
•
Θ
))
[cf.
the
notation
of
Corollary
2.18,
(i)]
which
has
the
property
that
any
isomorphism
∼
Π
•
X
→
Π
tp
X
maps
the
above
collection
of
classes
to
the
collection
of
classes
of
H
1
(
Ÿ
,
(l
·
Δ
Θ
))
given
by
some
multiple
of
the
collection
of
classes
η̈
Θ,l·Z×μ
2
by
an
l-th
root
of
unity
[cf.
Corollary
2.8,
(i)].
In
particular,
given
any
projective
system
of
mono-
theta
environments
∗
γ
M
,M
.
.
.
−→M
•
M
−→
M
•
M
−→
.
.
.
as
in
(ii),
by
taking
a
compatible
system
of
members
of
the
above
collections
of
classes
associated
to
the
[“Π
•
X
”
arising,
as
in
Corollary
2.18,
(iii),
from
the]
M
•
M
,
applying
the
isomorphisms
of
cyclotomes
of
(i),
and
adding
the
resulting
classes
to
the
[“theta”]
sections
[cf.
Definition
2.13,
(i),
(c)]
of
each
M
•
M
,
one
obtains
a
projective
system
of
bi-theta
environments
γ
M
,M
.
.
.
−→
B
•
M
−→
B
•
M
−→
.
.
.
that
is
isomorphic
to
some
“natural
projective
system
of
bi-theta
environ-
ments”
[of
standard
type]
β
M
,M
.
.
.
−→
B
M
−→
B
M
−→
.
.
.
60
SHINICHI
MOCHIZUKI
∼
[i.e.,
there
exist
isomorphisms
α
M
:
B
M
→
B
•
M
such
that
γ
M
,M
◦α
M
=
α
M
◦β
M
,M
,
for
M,
M
∈
E
satisfying
M
|M
]
as
in
Corollary
2.16.
Proof.
First,
we
consider
assertion
(i).
Observe
that
since
the
theta
and
algebraic
[i.e.,
“tautological”]
sections
coincide
over
Ker(Π
•
Y
(Π
•
Y
)
Θ
)
[cf.
Proposition
1.3],
it
follows
that
Ker(Π
•
Π
•
|
(Π
•
Y
)
Θ
)
may
be
constructed
as
the
image
via
the
theta
section
[cf.
Definition
2.13,
(ii),
(c)]
of
Ker(Π
•
Y
(Π
•
Y
)
Θ
).
The
subquotients
Π
•
|
(l·Δ
•
Θ
)
⊆
Π
•
|
(Δ
•
Y
)
Θ
⊆
Π
•
|
(Π
•
Y
)
Θ
may
then
be
constructed
as
the
inverse
images
via
the
resulting
quotient
Π
•
|
(Π
•
Y
)
Θ
(Π
•
Y
)
Θ
of
the
subquotients
(l
·
Δ
•
Θ
)
⊆
(Δ
•
Y
)
Θ
⊆
(Π
•
Y
)
Θ
of
Corollary
2.18,
(i).
The
splitting
of
the
natural
surjection
Π
•
|
(l·Δ
•
Θ
)
(l
·
Δ
•
Θ
)
corresponding
to
the
theta
section
may
then
be
obtained
directly
from
the
“theta
section
portion”
of
the
data
that
constitutes
a
mono-theta
environment
[cf.
Def-
inition
2.13,
(ii),
(c)];
the
splitting
corresponding
to
the
algebraic
section
[i.e.,
the
“tautological
section”]
may
then
be
constructed
via
the
algorithm
described
in
Proposition
2.14,
(i).
This
completes
the
proof
of
assertion
(i).
Assertion
(ii)
follows
immediately
from
Corollary
2.18,
(iv).
Here,
relative
to
the
point
of
view
of
Remark
2.16.1,
(iii),
we
note
that
assertion
(ii)
may
be
thought
of
as
a
conse-
quence
of
the
fact
that
[as
is
easily
verified]
the
“R
1
lim
←−
’s”
of
the
projective
system
“{Hom(Z/2Z,
Z/N
Z)}
N∈E
”
of
Corollary
2.18,
(iv),
as
well
as
the
projective
system
“{μ
N
}
N∈E
”
[cf.
the
superscripted
“μ’s”
of
Corollary
2.18,
(iv)],
vanish.
Finally,
we
consider
assertion
(iii).
An
algorithm
for
constructing
the
étale
theta
function
of
standard
type
is
described
in
the
proofs
of
Theorems
1.6,
1.10;
Corollary
2.8,
(i)
[cf.
also
the
proof
of
[Mzk14],
Theorem
6.8,
(iii)].
[Here,
we
recall
in
passing
that
this
“algorithm”
consists
essentially
of
restricting
[candidates
for]
the
étale
theta
function
to
the
decomposition
groups
of
certain
torsion
points.]
The
remainder
of
assertion
(iii)
follows,
in
light
of
the
cyclotomic
rigidity
of
assertion
(i)
and
the
discrete
rigidity
of
assertion
(ii),
from
the
construction
of
the
model
bi-theta
environment
in
the
discussion
preceding
Definition
2.13.
Remark
2.19.1.
One
way
to
try
to
eliminate
the
indeterminacy
discussed
in
Remarks
2.16.1,
2.16.2
is
to
attempt
to
work
with
profinite
coverings
of
X
log
that
correspond
to
the
covering
X
log
→
X
log
for
“l
infinite”.
On
the
other
hand,
such
coverings
amount
to
taking
N
-th
roots
[for
all
integers
N
≥
1]
of
the
theta
function.
In
particular,
when
N
is
a
power
of
p,
this
has
the
effect
of
annihilating
the
differ-
entials
of
the
curve
under
consideration.
Since
the
differentials
of
the
curve
play
an
essential
role
in
the
proof
of
the
main
result
of
[Mzk11],
it
thus
seems
unrealistic
[at
least
at
the
time
of
writing]
to
expect
to
generalize
the
main
result
of
[Mzk11]
[hence
also
the
theory
of
§1,
which
depends
on
this
result
of
[Mzk11]
in
an
essential
way]
so
as
to
apply
to
such
profinite
coverings.
THE
ÉTALE
THETA
FUNCTION
61
Remark
2.19.2.
The
“cyclotomic
rigidity”
of
Corollary
2.19,
(i),
is
a
consequence
of
the
theta
section
portion
of
the
data
that
constitutes
a
mono-theta
environment
[cf.
Definition
2.13,
(ii),
(c)],
together
with
the
subtle
property
of
the
commutator
[−,
−]
discussed
in
Proposition
2.12
[which
takes
the
place
of
the
algebraic
section,
an
object
which
does
not
appear
in
a
mono-theta
environment].
Note
that
this
subtle
property
depends
in
an
essential
way
on
the
fact
that
the
étale
theta
class
in
question
determines
an
isomorphism
between
the
subquotient
Δ
Θ
of
the
tem-
pered
fundamental
group
and
the
cyclotomic
coefficients
under
consideration
[cf.
Proposition
1.3].
In
particular:
This
subtle
property
fails
to
hold
if
instead
of
considering
η̈
Θ,l·Z×μ
2
over
log
Ÿ
—
i.e.,
the
first
power
of
an
l-th
root
of
the
theta
function
[cf.
the
discussion
preceding
Definition
2.7]
—
one
attempts
to
use
some
M
-th
power
of
the
l-th
root
of
the
theta
function
for
M
>
1.
Put
another
way,
if
one
tries
to
work
with
such
an
M
-th
power,
where
M
>
1,
then
one
ends
up
only
being
able
to
assert
the
desired
“cyclotomic
rigidity”
for
the
submodule
M
·μ
N
⊆
μ
N
[for,
say,
N
divisible
by
M
];
that
is
to
say,
the
“remainder”
of
μ
N
is
not
rigid,
but
rather
subject
to
an
indeterminacy
with
respect
to
the
action
of
Ker((Z/N
Z)
×
(Z/(N/M
)Z)
×
).
Alternatively,
if,
instead
of
working
with
torsion
coefficients
[i.e.,
μ
N
]
one
works
with
Z-flat
coefficients
[e.g.,
the
inverse
limit
of
the
μ
N
,
as
N
ranges
over
the
integers
≥
1],
then
one
may
still
obtain
the
[
Z-flat
analogue
of
the]
desired
“cyclotomic
rigidity”
property
of
Corollary
2.19,
(i),
for
M
>
1,
but
only
at
the
cost
of
working
with
“profinite
coverings”
whose
finite
subcoverings
are
“immune”
to
automorphism
indeterminacy,
which
[cf.
Corollary
5.12
and
the
following
remarks
in
§5
below]
appears
to
be
somewhat
unnatural.
Remark
2.19.3.
In
the
context
of
the
projective
systems
discussed
in
Corollary
2.19,
(ii),
(iii),
if
one
writes
Δ
[μ
∞
]
for
the
inverse
limit
of
the
Δ
[μ
N
]
[as
N
ranges
over
the
integers
≥
1],
then
one
may
think
of
the
isomorphism
∼
(l
·
Δ
Θ
)
→
Δ
[μ
∞
]
arising
from
the
“cyclotomic
rigidity”
[i.e.,
the
compatible
isomorphisms
of
cyclo-
tomes]
of
Corollary
2.19,
(i),
as
determining
a
sort
of
“integral
structure”,
i.e.,
a
sort
of
“basepoint”
corresponding
to
the
first
power
of
the
l-th
root
of
the
theta
function,
relative
to
the
various
M
-th
powers
of
the
l-th
root
of
the
theta
func-
tion
[cf.
Remark
2.19.2]
obtained
by
composing
this
isomorphism
with
the
map
Δ
[μ
∞
]
→
Δ
[μ
∞
]
on
Δ
[μ
∞
]
given
by
multiplication
by
M
.
Put
another
way:
To
work
in
the
absence
of
such
a
“basepoint”
amounts
to
sacrificing
the
datum
of
an
intrisically
defined
“origin”,
or
“fixed
reference
point”,
in
the
system
...
M
·
−→
Δ
[μ
∞
]
M
·
−→
Δ
[μ
∞
]
M
·
−→
Δ
[μ
∞
]
M
·
−→
...
62
SHINICHI
MOCHIZUKI
obtained
by
multiplication
by
M
on
the
cyclotome
Δ
[μ
∞
]
.
Put
another
way,
there
is
no
intrinsic
way
to
distinguish
“Δ
[μ
∞
]
”
from
“M
·
Δ
[μ
∞
]
”
—
i.e.,
the
distinction
between
these
two
objects
is
entirely
a
matter
of
“arbitrary
labels”
[which
are
typically
implicit
in
classical
discussions
of
arithmetic
geometry
—
cf.
the
discussion
of
the
Introduction
to
the
present
paper].
Remark
2.19.4.
Before
proceeding,
it
is
natural
to
pause
and
reflect
on
the
topic
of
precisely
what
one
gains
from
the
discrete
and
cyclotomic
rigidity
of
Corollary
2.19,
(i),
(ii).
On
the
one
hand,
discrete
rigidity
assures
one
that,
when
one
works
with
the
projective
systems
discussed
in
Corollary
2.19,
(ii),
(iii),
one
may
restrict
to
the
Z-translates
of
[an
l-th
root
of]
the
theta
function
without
having
to
worry
which
are
“unnatural”.
At
the
level
of
about
confusion
with
arbitrary
Z-translates,
theta
values
[cf.,
e.g.,
Proposition
1.4,
(iii);
the
labels
of
Corollary
2.9],
this
means
that
one
obtains
values
in
K
×
,
as
opposed
to
(K
×
)
∧
;
in
particular,
it
makes
sense
to
perform
[not
just
multiplication
operations,
but
also]
addition
operations
involving
these
values
in
K
×
⊆
K,
which
is
not
possible
with
arbitrary
elements
of
(K
×
)
∧
.
On
the
other
hand,
cyclotomic
rigidity
assures
one
that
one
may
work
with
the
first
power
of
[an
l-th
root
of]
the
theta
function
without
having
to
worry
that
this
×
.
At
first
power
might
be
“confused
with
some
arbitrary
λ-th
power”,
for
λ
∈
Z
the
level
of
theta
values
[cf.,
e.g.,
Proposition
1.4,
(iii);
the
labels
of
Corollary
2.9],
this
means
that
one
need
not
worry
about
confusion
between
the
“original
desired
values”
in
K
×
⊆
(K
×
)
∧
and
arbitrary
λ-th
powers
of
such
values
in
(K
×
)
∧
,
for
×
—
where
again
it
is
useful
to
recall
that
raising
to
the
λ-th
power
on
(K
×
)
∧
λ
∈
Z
×
]
is
not
a
ring
homomorphism
[i.e.,
not
compatible
with
addition]
unless
[for
λ
∈
Z
λ
=
1.
Remark
2.19.5.
Recall
that
in
the
proof
of
[Mzk13],
Theorem
4.3
[cf.
especially
the
proof
of
[Mzk2],
Lemma
2.5,
(ii)],
one
finds
a
“group-theoretic
algorithm”
for
constructing
a
certain
natural
isomorphism
of
cyclotomes,
between
one
cyclotome
of
geometric
origin
—
which,
in
the
situation
of
Corollaries
2.18,
2.19,
essentially
amounts
to
(l
·
Δ
Θ
)
—
and
one
cyclotome
of
arithmetic
origin
—
which,
in
the
situation
of
Corollaries
2.18,
2.19,
arises
from
G
K
.
If
one
combines
this
isomorphism
of
cyclotomes
with
the
isomorphism
of
cyclotomes
given
in
Corollary
2.19,
(i),
the
resulting
“two-layer
isomorphism
of
cyclotomes
structure”
is
reminiscent
of
the
“Griffiths
semi-transversality”
of
the
“crystalline
theta
object”
in
the
Hodge-
Arakelov
theory
of
elliptic
curves
[cf.
[Mzk5],
Theorem
2.8],
which
arises
from
the
“two-layer
deformation”
that
occurs
in
the
consideration
of
the
“crystalline
theta
object”
[i.e.,
a
deformation
of
the
elliptic
curve,
together
with
a
deformation
of
an
ample
line
bundle
on
the
deformed
elliptic
curve].
THE
ÉTALE
THETA
FUNCTION
63
Section
3:
Tempered
Frobenioids
In
the
present
§3,
we
construct
certain
Frobenioids
[cf.
the
theory
of
[Mzk17],
[Mzk18]]
arising
from
the
geometry
of
line
bundles
on
tempered
coverings
of
a
p-adic
curve.
After
discussing
various
basic
properties
of
these
“tempered
Frobenioids”
[cf.
Theorem
3.7;
Corollary
3.8],
we
explain
how
certain
aspects
of
the
theory
of
the
étale
theta
function
discussed
in
§1,
§2
may
be
interpreted
from
the
point
of
view
of
tempered
Frobenioids
[cf.
Example
3.9].
Let
L
be
a
finite
extension
of
Q
p
[where
p
is
a
prime
number],
with
ring
of
integers
O
L
and
residue
field
k
L
;
T
the
formal
scheme
given
by
the
p-adic
completion
of
Spec(O
L
);
T
log
the
formal
log
scheme
obtained
by
equipping
T
with
the
log
structure
determined
by
the
unique
closed
point
of
Spec(O
L
);
Z
log
a
stable
log
curve
over
T
log
.
Also,
we
assume
that
the
special
fiber
Z
k
L
of
Z
is
split,
and
that
the
generic
def
fiber
of
the
algebrization
of
Z
log
is
a
smooth
log
curve.
Write
Z
log
=
Z
log
×
O
L
L
for
the
ringed
space
with
log
structure
obtained
by
tensoring
the
structure
sheaf
of
Z
over
O
L
with
L.
In
the
following
discussion,
we
shall
often
[by
abuse
of
notation]
use
the
notation
Z
log
also
to
denote
the
generic
fiber
of
the
algebrization
of
Z
log
[cf.
§1].
log
deter-
The
universal
covering
of
the
dual
graph
of
the
special
fiber
Z
log
k
L
of
Z
mines
an
infinite
Galois
étale
covering
log
Z
log
∞
→
Z
of
Z
log
;
such
“universal
combinatorial
coverings”
appear
in
the
theory
of
the
tem-
pered
fundamental
group
[cf.
[André],
§4;
[Mzk14],
Example
3.10].
Thus,
Z
log
∞
is
a
def
log
formal
log
scheme;
write
Z
∞
=
Z
log
∞
×
O
L
L.
Also,
we
shall
refer
to
the
inverse
log
image
of
the
divisor
of
cusps
of
Z
log
in
Z
log
∞
as
the
divisor
of
cusps
of
Z
∞
and
to
log
Z
log
∞
as
the
stable
model
of
Z
∞
.
Definition
3.1.
(i)
A
divisor
on
Z
∞
whose
support
lies
in
the
special
fiber
(Z
∞
)
k
L
(respectively,
log
the
divisor
of
cusps
of
Z
log
∞
;
the
union
of
the
special
fiber
and
divisor
of
cusps
of
Z
∞
)
will
be
referred
to
as
a
non-cuspidal
log-divisor
(respectively,
cuspidal
log-divisor;
log-divisor)
on
Z
log
∞
.
Write
log
log
log
DIV(Z
log
∞
)
(respectively,
DIV
+
(Z
∞
);
Div(Z
∞
);
Div
+
(Z
∞
))
for
the
monoid
of
log-divisors
(respectively,
effective
log-divisors;
Cartier
log-divisors;
effective
Cartier
log-divisors)
on
Z
log
.
Thus,
we
have
natural
inclusions
log
log
Div
+
(Z
log
∞
)
⊆
DIV
+
(Z
∞
)
⊆
DIV(Z
∞
)
log
log
Div
+
(Z
log
∞
)
⊆
Div(Z
∞
)
⊆
DIV(Z
∞
)
log
gp
and
a
natural
identification
DIV(Z
log
∞
)
=
DIV
+
(Z
∞
)
.
64
SHINICHI
MOCHIZUKI
(ii)
A
nonzero
meromorphic
function
on
Z
log
∞
whose
divisor
of
zeroes
and
poles
is
a
log-divisor
will
be
referred
to
as
a
log-meromorphic
function
on
Z
log
∞
.
The
group
of
log
log
log-meromorphic
functions
on
Z
∞
will
be
denoted
Mero(Z
∞
).
A
log-meromorphic
function
arising
from
L
×
will
be
referred
to
as
constant.
Proposition
3.2.
(Divisors
and
Rational
Functions
on
Universal
Com-
binatorial
Coverings)
In
the
notation
of
the
above
discussion:
log
(i)
There
exists
a
positive
integer
n
such
that
n
·
DIV
+
(Z
log
∞
)
⊆
Div
+
(Z
∞
),
log
n
·
DIV(Z
log
∞
)
⊆
Div(Z
∞
).
In
particular,
there
exists
a
natural
isomorphism
∼
pf
pf
Div
+
(Z
log
→
DIV
+
(Z
log
∞
)
∞
)
pf
—
where
DIV
+
(Z
log
∞
)
may
be
naturally
identified
with
a
direct
product
of
copies
of
Q
≥0
,
indexed
by
the
cusps
[i.e.,
irreducible
components
of
the
divisor
of
cusps]
and
irreducible
components
of
the
special
fiber
of
Z
log
∞
.
log
(ii)
The
structure
morphism
Z
log
determines
a
natural
isomorphism
∞
→
T
O
L
→
Γ(Z
∞
,
O
Z
∞
)
—
i.e.,
“all
regular
functions
on
Z
∞
are
constant”.
∼
(iii)
Let
f
be
a
nonzero
meromorphic
function
on
Z
∞
such
that
for
every
N
N
∈
N
≥1
[cf.
§0],
there
exists
a
meromorphic
function
g
N
on
Z
∞
such
that
g
N
=
f
.
Then
f
=
1.
Proof.
To
verify
assertion
(i),
let
us
first
observe
that
the
completion
of
Z
∞
along
a
node
of
Z
∞
may
be
identified
with
the
formal
spectrum
of
a
complete
local
ring
of
e
the
form
O
L
[[x,
y]]/(xy
−
π
L
),
where
π
L
is
a
uniformizer
of
O
L
,
and
e
is
a
positive
integer;
moreover,
despite
the
“infinite”
nature
of
Z
∞
,
the
number
of
“e’s”
that
occur
at
completions
of
Z
∞
along
its
nodes
is
finite
[cf.
the
definition
of
Z
log
∞
in
log
terms
of
Z
!].
Now
assertion
(i)
follows
from
the
fact
that
the
two
irreducible
components
of
the
special
fiber
of
this
formal
spectrum
determine
divisors
D,
E
such
that
e
·
D,
e
·
E
are
Cartier
[i.e.,
since
they
occur
as
the
schematic
zero
loci
of
“x”,
“y”].
Next,
we
consider
assertion
(ii).
Let
0
=
f
∈
Γ(Z
∞
,
O
Z
∞
);
write
V
(f
)
for
the
schematic
zero
locus
of
f
on
Z
∞
.
Now
observe
that
for
each
irreducible
component
C
of
(Z
∞
)
k
L
,
there
exists
an
e
C
∈
Z
≥0
such
that
the
meromorphic
function
−e
C
f
·
π
L
—
where
π
L
is
a
uniformizer
of
O
L
—
has
no
zeroes
or
poles
at
the
generic
point
of
C.
By
the
discrete
structure
of
Z
≥0
,
it
follows
that
there
exists
an
irreducible
component
C
1
such
that
e
C
1
≤
e
C
,
for
all
irreducible
components
C
of
(Z
∞
)
k
L
.
def
−e
Thus,
the
meromorphic
function
f
1
=
f
·
π
L
C
1
is
regular,
i.e.,
f
1
∈
Γ(Z
∞
,
O
Z
∞
),
and,
moreover,
has
nonzero
restriction
to
(Z
∞
)
k
L
.
On
the
other
hand,
since
(Z
∞
)
k
L
is
connected
and
reduced,
and
each
irreducible
component
C
of
(Z
∞
)
k
L
is
proper
and
geometrically
integral
over
k
L
[since
we
assumed
that
Z
k
L
of
Z
is
split],
it
THE
ÉTALE
THETA
FUNCTION
65
follows
that
immediately
that
the
natural
morphism
k
L
→
Γ((Z
∞
)
k
L
,
O
(Z
∞
)
kL
)
is
×
,
g
∈
Γ(Z
∞
,
O
Z
∞
).
Thus,
an
isomorphism,
hence
that
f
1
=
λ
+
π
L
·
g,
where
λ
∈
O
L
by
repeating
this
argument
[with
“f
”
replaced
by
“g”]
and
applying
the
p-adic
completeness
of
Z
∞
,
we
conclude
that
f
∈
O
L
,
as
desired.
Finally,
we
consider
assertion
(iii).
Since
Z
∞
is
locally
noetherian,
it
follows
immediately
from
the
existence
of
the
g
N
that
the
divisor
of
zeroes
and
poles
of
f
×
.
Since
L
is
a
finite
extension
is
0,
hence,
by
assertion
(ii),
that
f
is
a
constant
∈
O
L
×
of
Q
p
,
it
thus
follows
from
the
well-known
structure
of
O
L
that
×
N
f
∈
(O
L
)
=
{1}
N∈N
≥1
—
i.e.,
that
f
=
1,
as
desired.
Next,
let
K
be
a
finite
extension
of
Q
p
,
with
ring
of
integers
O
K
and
residue
field
k;
K
a
finite
Galois
extension
of
K
[cf.
Remark
3.3.2
below],
with
ring
of
integers
O
K
;
S
the
formal
stack
given
by
forming
the
stack-theoretic
quotient
with
respect
to
the
natural
action
of
Gal(K
/K)
of
the
p-adic
completion
of
Spec(O
K
);
S
log
the
formal
log
stack
obtained
by
equipping
S
with
the
log
structure
deter-
mined
by
the
unique
closed
point
of
Spec(O
K
);
X
log
a
stable
log
orbicurve
[cf.
§0]
def
over
S
log
.
Also,
we
assume
that
the
generic
fiber
X
log
=
X
log
×
O
K
K
[of
the
algebrization]
of
X
log
is
a
smooth
log
orbicurve
[cf.
§0].
Write
B
temp
(X
log
)
for
the
temperoid
of
tempered
coverings
of
X
log
[cf.
[Mzk14],
Example
3.10],
B(Spec(K))
for
the
Galois
category
of
finite
étale
coverings
of
Spec(K),
and
def
def
D
0
=
B
temp
(X
log
)
0
;
D
cnst
=
B(Spec(K))
0
—
where
the
superscript
“0”
denotes
the
full
subcategory
constituted
by
the
con-
def
temp
(X
log
)
is
nected
objects
[cf.
[Mzk17],
§0,
for
more
details].
Thus,
if
Π
tp
X
=
π
1
log
[cf.
[André],
§4;
[Mzk14],
Example
3.10],
the
tempered
fundamental
group
of
X
temp
log
(X
)
is
naturally
isomorphic
[as
a
temperoid]
to
the
then
the
temperoid
B
tp
temp
temperoid
B
(Π
X
)
associated
to
the
tempered
group
Π
tp
X
[cf.
§0].
In
a
similar
vein,
the
Galois
category
B(Spec(K))
is
naturally
equivalent
to
the
Galois
category
B(G
K
)
associated
to
the
absolute
Galois
group
G
K
of
K.
Also,
we
observe
that
the
natural
surjection
Π
tp
X
G
K
determines
a
natural
functor
D
0
→
D
cnst
[cf.
def
tp
[Mzk18],
Example
1.3,
(ii)].
Write
Δ
tp
X
=
Ker(Π
X
G
K
).
Definition
3.3.
(i)
Let
Δ
be
a
tempered
group
[cf.
§0].
Then
we
shall
refer
to
as
a
tempered
filter
on
Δ
a
countable
collection
of
characteristic
open
subgroups
of
finite
index
Δ
fil
=
{Δ
fil
i
}
i∈I
66
SHINICHI
MOCHIZUKI
of
Δ
such
that
the
following
conditions
are
satisfied:
(a)
We
have:
i∈I
Δ
fil
i
=
{1}.
fil,∞
(b)
Every
Δ
fil
[which
is
i
admits
a
minimal
co-free
subgroup
[cf.
§0]
Δ
i
necessarily
characteristic
as
a
subgroup
of
Δ].
(c)
For
each
open
subgroup
H
⊆
Δ,
there
exists
a
[necessarily
unique]
i
H
∈
I
⊆
H,
and,
moreover,
for
every
i
∈
I,
Δ
fil,∞
⊆
H
implies
such
that
Δ
fil,∞
i
H
i
fil,∞
fil,∞
⊆
Δ
i
H
.
Δ
i
In
the
situation
of
(c),
we
shall
refer
to
Δ
fil,∞
as
the
Δ
fil
-closure
of
H
in
Δ.
i
H
log
(ii)
We
shall
refer
to
a
tempered
filter
on
Δ
tp
.
Let
X
as
a
tempered
filter
on
X
Δ
fil
=
{Δ
fil
i
}
i∈I
be
a
tempered
filter
on
X
log
.
Suppose
that
Z
log
→
X
log
is
a
finite
étale
Galois
covering
that
admits
a
stable
model
Z
log
over
the
ring
of
integers
of
the
extension
field
of
K
determined
by
the
integral
closure
of
K
in
Z
log
such
that
the
special
fiber
of
Z
log
is
split
[i.e.,
Z
log
is
a
curve
as
in
the
discussion
at
the
beginning
of
the
present
§3],
and,
moreover,
the
open
subgroup
determined
by
the
[geometric
tp
log
log
portion
of]
this
covering
is
equal
to
one
of
the
Δ
fil
i
⊆
Δ
X
.
Write
Z
∞
→
Z
log
for
the
“universal
combinatorial
covering”
of
Z
log
and
Z
∞
→
Z
log
for
the
generic
log
log
[so
Z
∞
→
Z
log
corresponds
to
the
subgroup
Δ
fil,∞
⊆
Δ
tp
fiber
of
Z
log
∞
→
Z
i
X
—
cf.
[André],
Proposition
4.3.1;
[André],
the
proof
of
Lemma
6.1.1].
Then
we
shall
log
refer
to
Z
∞
→
X
log
as
a
Δ
fil
-covering
of
X
log
.
If,
moreover,
Y
log
→
X
log
is
a
connected
tempered
covering,
which
determines
an
open
subgroup
H
⊆
Δ
tp
X
,
and
fil,∞
fil
log
⊆
H
is
the
Δ
-closure
of
H,
then
we
shall
refer
to
any
covering
Z
∞
→
Y
log
Δ
i
log
whose
composite
with
Y
log
→
X
log
is
the
covering
Z
∞
→
X
log
as
a
Δ
fil
-closure
of
Y
log
→
X
log
.
[Thus,
the
geometric
portion
—
but
not
the
base
field!
—
of
a
Δ
fil
-closure
of
Y
log
→
X
log
is
uniquely
determined
up
to
isomorphism.]
log
.
Then
for
any
connected
(iii)
Let
Δ
fil
=
{Δ
fil
i
}
i∈I
be
a
tempered
filter
on
X
log
log
→
X
,
it
makes
sense
to
define
tempered
covering
Y
def
log
log
def
log
log
log
Gal(Z
∞
/Y
Φ
0
(Y
log
)
=
−
lim
log
Div
+
(Z
∞
)
→
Z
∞
log
Gal(Z
∞
/Y
B
0
(Y
log
)
=
−
lim
log
Mero(Z
∞
)
→
Z
∞
)
)
log
—
where
the
inductive
limits
range
over
the
Δ
fil
-closures
Z
∞
→
Y
log
of
Y
log
→
log
X
;
the
superscript
Galois
groups
denote
the
submonoids
of
elements
fixed
by
the
Galois
group
in
question.
Moreover,
by
(i),
(c),
the
assignments
Y
log
→
Φ
0
(Y
log
),
Y
log
→
B
0
(Y
log
)
determine
functors
Φ
0
:
D
0
→
Mon;
B
0
:
D
0
→
Mon
THE
ÉTALE
THETA
FUNCTION
67
—
where
“Mon”
is
the
category
of
commutative
monoids
[cf.
[Mzk17],
§0]
—
together
with
a
natural
transformation
B
0
→
Φ
gp
0
[given
by
assigning
to
a
log-meromorphic
function
its
log-divisor
of
zeroes
and
poles],
whose
image
we
denote
by
Φ
birat
⊆
Φ
gp
0
0
.
Also,
we
shall
write
F
0
⊆
B
0
for
the
subfunctor
determined
by
the
constant
log-meromorphic
functions
and
Φ
cnst
⊆
Φ
gp
0
0
gp
for
the
image
of
F
0
in
Φ
0
.
Remark
3.3.1.
Note
that
the
set
of
primes
[cf.
[Mzk17],
§0]
of
the
monoid
log
Gal(Z
∞
/Y
Div
+
(Z
log
∞
)
log
)
appearing
in
the
definition
of
Φ
0
(Y
log
)
is
in
natural
bijective
correspondence
with
log
the
set
of
Gal(Z
∞
/Y
log
)-orbits
of
prime
log-divisors
on
Z
log
∞
[cf.
Proposition
3.2,
log
→
Y
log
differ
only
(i)].
Moreover,
since,
by
definition,
different
Δ
fil
-closures
Z
∞
by
an
extension
of
the
base
field
K,
it
follows
immediately
that
in
the
inductive
limit
appearing
in
the
definition
of
Φ
0
(Y
log
),
the
maps
between
monoids
induce
isomorphisms
of
monoids
on
the
respective
perfections,
hence
that
the
resulting
sets
of
primes
map
bijectively
to
one
another.
Remark
3.3.2.
Note
that
by
taking
the
extension
field
K
used
to
define
the
stack
structure
of
S
to
be
“sufficiently
large”,
one
may
treat
the
case
in
which
X
log
fails
to
have
stable
reduction
over
O
K
.
Moreover,
although
at
first
sight
the
choice
of
K
may
appear
to
be
somewhat
arbitrary,
one
verifies
immediately
that
the
category
D
0
,
as
well
as
the
monoids
Φ
0
,
B
0
on
D
0
,
are
unaffected
by
replacing
K
by
some
larger
finite
Galois
extension
of
K.
Proposition
3.4.
(Divisor
and
Rational
Function
Monoids)
In
the
nota-
tion
of
the
above
discussion:
(i)
Φ
0
(Y
log
),
as
well
as
each
of
the
monoids
log
Gal(Z
∞
/Y
Div
+
(Z
log
∞
)
log
)
appearing
in
the
inductive
limit
defining
Φ
0
(Y
log
),
is
perf-factorial
[cf.
[Mzk17],
Definition
2.4,
(i)].
Moreover,
every
endomorphism
of
Φ
0
(Y
log
)
or
one
of
the
log
Gal(Z
∞
/Y
log
)
induced
by
an
endomorphism
of
Y
log
over
X
log
is
non-
Div
+
(Z
log
∞
)
dilating
[cf.
[Mzk17],
Definition
1.1,
(i)].
In
particular,
the
functor
Φ
0
defines
a
divisorial
monoid
[cf.
[Mzk17],
Definition
1.1,
(i),
(ii)]
on
D
0
which
is,
moreover,
perf-factorial
and
non-dilating.
(ii)
Suppose
that
Y
log
→
X
log
is
a
connected
tempered
covering
such
that
the
composite
morphism
Y
log
→
Spec(K)
factors
through
Spec(L),
for
some
finite
68
SHINICHI
MOCHIZUKI
extension
L
of
K,
in
such
a
way
that
Y
log
is
geometrically
connected
over
L.
Then
we
have
natural
isomorphisms
of
monoids
∼
×
log
→
Ker(B
0
(Y
log
)
→
Φ
gp
))
⊆
B
0
(Y
log
)
O
L
0
(Y
∼
log
O
L
→
B
0
(Y
log
)
×
Φ
gp
log
)
Φ
0
(Y
);
0
(Y
∼
L
×
→
F
0
(Y
log
)
⊆
B
0
(Y
log
)
”
is
as
in
[Mzk18],
Example
1.1.
—
where
“O
L
Proof.
First,
we
consider
assertion
(i).
Let
M
be
one
of
the
monoids
under
consideration.
The
fact
that
M
is
divisorial
is
immediate
from
the
definitions.
The
fact
that
M
is
perf-factorial
then
follows
immediately
from
Proposition
3.2,
(i)
[cf.
also
the
description
of
the
primes
of
M
in
terms
of
“orbits
of
prime
log-divisors”
given
in
Remark
3.3.1].
Now
let
α
be
an
endomorphism
of
M
that
is
induced
by
an
endomorphism
of
Y
log
over
X
log
such
that
α
induces
the
identity
endomorphism
on
the
set
of
primes
of
M
.
Then
by
considering
local
functions
on
Z
∞
that
arise
from
local
functions
on
X
and
vanish
at
various
primes
of
M
,
it
follows
that
α
is
the
identity,
as
desired.
This
completes
the
proof
of
assertion
(i).
Assertion
(ii)
follows
immediately
from
Proposition
3.2,
(ii)
[and
the
definitions].
Lemma
3.5.
(Perfections
and
Realifications
of
Perf-factorial
Sub-
monoids)
Let
P
,
Q
be
perf-factorial
monoids
such
that:
(a)
P
is
a
submonoid
of
Q;
(b)
P
is
group-saturated
[cf.
§0]
in
Q;
(c)
R
supports
Q
[cf.
[Mzk17],
Definition
2.4,
(ii)].
Then:
(i)
The
inclusion
P
→
Q
extends
uniquely
to
inclusions
P
pf
→
Q,
P
rlf
→
Q.
(ii)
Relative
to
the
inclusions
of
(i),
P
pf
,
P
rlf
are
group-saturated
in
Q.
Proof.
Indeed,
the
portion
of
assertions
(i),
(ii)
involving
“P
pf
”
follows
imme-
diately
from
the
definitions.
Next,
let
p
∈
Prime(P
)
[where
“Prime(−)”
is
as
in
[Mzk17],
§0].
Since
P
is
perf-factorial,
it
follows
that
the
“primary
component”
P
p
associated
to
p
is
isomorphic
to
Z
≥0
,
Q
≥0
,
or
R
≥0
[cf.
[Mzk17],
Definition
2.4,
(i),
(b)].
Since
R
≥0
acts
on
Q
[cf.
condition
(c)],
it
thus
follows
that
the
natural
homomorphism
of
monoids
P
p
→
P
→
Q
extends
[uniquely]
to
a
homomorphism
of
monoids
P
p
rlf
→
Q.
Next,
observe
that
it
follows
from
the
definition
of
the
re-
alification
[cf.
[Mzk17],
Definition
2.4,
(i)]
that
for
every
a
∈
P
rlf
,
there
exists
an
a
∈
P
pf
such
that
a
≥
a.
In
particular,
it
follows
that
for
each
q
∈
Prime(Q),
the
sum
of
the
images
of
the
various
“primary
components
a
p
∈
P
p
of
a”
[as
p
ranges
over
the
elements
of
Prime(P
)]
in
Q
q
∼
=
R
≥0
is
bounded
above
[i.e.,
by
the
image
in
R
Q
q
∼
of
a
,
which
is
well-defined
since
a
∈
P
pf
].
Thus,
this
sum
converges
to
=
≥0
an
element
of
Q
q
∼
=
R
≥0
.
Now,
letting
q
range
over
the
elements
of
Prime(Q),
we
conclude
that
we
obtain
a
homomorphism
of
monoids
rlf
P
rlf
→
Q
pf
factor
=
Q
factor
[relative
to
the
notation
of
[Mzk17],
Definition
2.4,
(i),
(c)].
Since,
moreover,
Q
is
perf-factorial,
it
follows
from
[Mzk17],
Definition
2.4,
(i),
(d)
[together
with
the
THE
ÉTALE
THETA
FUNCTION
69
existence
of
an
a
∈
P
pf
such
that
a
≥
a],
that
this
homomorphism
factors
through
Q,
hence
determines
a
homomorphism
of
monoids
φ
:
P
rlf
→
Q
that
is
[easily
verified
to
be]
uniquely
characterized
by
the
property
that
it
extends
the
natural
homomorphism
of
monoids
P
pf
→
Q.
Write
φ
gp
:
(P
rlf
)
gp
→
Q
gp
for
the
induced
homomorphism
on
groupifications.
Next,
let
a,
b
∈
P
rlf
be
such
that
φ
gp
(a
−
b)
≥
0
[i.e.,
φ
gp
(a
−
b)
∈
Q].
Then
for
any
a
,
b
∈
P
pf
such
that
a
≥
a,
b
≤
b,
we
obtain
that
φ
gp
(a
−
b
)
≥
φ
gp
(a
−
b)
≥
0,
hence
[by
the
portion
of
assertion
(ii)
concerning
“P
pf
”]
that
a
≥
b
.
On
the
other
hand,
since
P
is
perf-
factorial
[cf.
[Mzk17],
Definition
2.4,
(i),
(d)],
it
follows
immediately
that
if
a
≥
b,
then
there
exist
a
,
b
∈
P
pf
(A)
such
that
a
≥
a,
b
≤
b,
a
≥
b
.
Thus,
we
conclude
that
a
≥
b.
In
particular,
if
φ
gp
(a
−
b)
=
0,
then
it
follows
that
there
exists
a
def
c
∈
P
rlf
[i.e.,
c
=
a
−
b]
such
that
φ(c)
=
0.
On
the
other
hand,
if
c
=
0,
then
[cf.
[Mzk17],
Definition
2.4,
(i),
(d)]
there
exists
a
c
∈
P
pf
such
that
0
<
c
≤
c,
hence
that
0
≤
φ(c
)
≤
0,
so
φ(c
)
=
0,
in
contradiction
to
the
injectivity
of
the
natural
homomorphism
of
monoids
P
pf
→
Q.
Thus,
we
conclude
that
φ
is
injective.
This
completes
the
proof
of
the
portion
of
assertions
(i),
(ii)
involving
“P
rlf
”.
Remark
3.5.1.
Observe
that
it
follows
immediately
from
Lemma
3.5,
(i),
that
a
nonzero
submonoid
P
of
an
R-monoprime
[cf.
[Mzk17],
§0]
monoid
Q
is
perf-
factorial
and
group-saturated
if
and
only
if
it
is
monoprime.
Remark
3.5.2.
Note
that
the
injectivity
portion
of
Lemma
3.5,
(i),
fails
to
hold
if
one
omits
the
crucial
hypothesis
that
P
is
group-saturated
in
Q.
Indeed,
this
may
def
def
be
seen,
for
instance,
by
considering
an
injection
P
=
Z
≥0
⊕
Z
≥0
→
Q
=
R
≥0
that
sends
the
elements
(1,
0);
(0,
1)
of
P
to
[nonzero]
Q-linearly
independent
elements
of
R
≥0
.
Definition
3.6.
In
the
notation
of
Definition
3.3,
(iii):
Λ
Λ
(i)
Let
Λ
be
a
monoid
type.
Define
Φ
Λ
0
,
B
0
,
F
0
as
follows:
def
Φ
Z
0
=
Φ
0
;
pf
Φ
Q
0
=
Φ
0
;
def
def
rlf
Φ
R
0
=
Φ
0
B
Z
0
=
B
0
;
def
pf
B
Q
0
=
B
0
;
def
birat
gp
B
R
⊆
(Φ
R
0
=
R
·
Φ
0
0
)
def
pf
F
Q
0
=
F
0
;
cnst
gp
F
R
⊆
(Φ
R
0
=
R
·
Φ
0
0
)
F
Z
0
=
F
0
;
def
def
def
—
where
Φ
rlf
0
is
as
in
[Mzk17],
Definition
2.4,
(i)
[cf.
Proposition
3.4,
(i)].
(ii)
Let
D
be
a
connected,
totally
epimorphic
category,
equipped
with
a
functor
D
→
D
0
;
def
Φ
⊆
Φ
R-log
=
Φ
R
0
|
D
a
group-saturated
[i.e.,
Φ(A)
is
group-saturated
in
Φ
R-log
(A),
∀A
∈
Ob(D)]
sub-
functor
in
monoids
which
determines
a
perf-factorial
divisorial
monoid
on
D
such
70
SHINICHI
MOCHIZUKI
that
the
following
conditions
are
satisfied:
(a)
the
[necessarily
group-saturated]
sub-
monoid
def
)|
D
×
(Φ
R-log
)
gp
Φ
⊆
Φ
R-log
Φ
bs-fld
=
(R
·
Φ
cnst
0
on
D
is
monoprime
[cf.
[Mzk17],
§0];
(b)
the
image
of
the
resulting
homomorphism
of
group-like
monoids
on
D
gp
→
(Φ
bs-fld
)
gp
=
(R
·
Φ
cnst
)|
D
×
(Φ
R-log
)
gp
Φ
gp
⊆
(Φ
R-log
)
gp
F
=
F
Λ
0
|
D
×
(Φ
R-log
)
gp
Φ
0
def
determines
a
subfunctor
in
nonzero
monoids
of
(Φ
bs-fld
)
gp
[i.e.,
for
every
A
∈
Ob(D),
the
homomorphism
F(A)
→
(Φ
bs-fld
)
gp
(A)
is
nonzero].
[Thus,
it
follows
from
these
conditions
that
for
every
A
∈
Ob(D),
the
image
of
the
homomorphism
F(A)
→
def
(Φ
bs-fld
)
gp
(A)
contains
a
nonzero
element
of
Φ
bs-fld
(A).]
Write
B
=
B
Λ
0
|
D
×
(Φ
R-log
)
gp
Φ
gp
→
Φ
gp
.
Thus,
the
data
(D,
Φ,
B,
B
→
Φ
gp
)
determines
a
model
Frobenioid
C
[cf.
[Mzk17],
Theorem
5.2,
(ii)].
We
shall
refer
to
a
Frobenioid
C
obtained
in
this
way
as
a
tempered
Frobenioid
and
to
Λ
as
the
monoid
type
of
the
tempered
Frobenioid
C.
If
C
is
of
rational
(respectively,
strictly
rational)
type
[a
property
which
is
completely
determined
by
Φ
—
cf.
[Mzk17],
Definition
4.5,
(ii)],
then
we
shall
say
that
Φ
is
rational
(respectively,
strictly
rational).
(iii)
If
A
∈
Ob(D),
then
we
shall
say
that
an
element
of
Φ(A)
is
non-cuspidal
(respectively,
cuspidal)
if
it
arises
[cf.
the
inductive
limit
that
appears
in
the
defi-
nition
of
Φ
0
]
from
a
non-cuspidal
(respectively,
cuspidal)
log-divisor;
we
shall
say
that
a
prime
p
of
the
monoid
Φ(A)
is
non-cuspidal
(respectively,
cuspidal)
if
the
primary
elements
of
Φ(A)
that
are
contained
in
p
are
non-cuspidal
(respectively,
cuspidal).
In
the
following,
we
shall
write
Φ(A)
ncsp
⊆
Φ(A);
Prime(Φ(A))
ncsp
⊆
Prime(Φ(A));
Φ(A)
csp
⊆
Φ(A)
Prime(Φ(A))
csp
⊆
Prime(Φ(A))
for
the
submonoids
of
non-cuspidal
and
cuspidal
elements
and
the
subsets
of
non-
cuspidal
and
cuspidal
primes,
respectively.
We
shall
refer
to
a
pre-step
of
C
as
non-cuspidal
(respectively,
cuspidal)
if
its
zero
divisor
is
non-cuspidal
(respectively,
cuspidal).
(iv)
The
data
[cf.
(ii)]
(D,
Φ
bs-fld
,
F,
F
→
(Φ
bs-fld
)
gp
)
determines
a
model
Frobenioid
C
bs-fld
THE
ÉTALE
THETA
FUNCTION
71
[cf.
[Mzk17],
Theorem
5.2,
(ii)].
Moreover,
the
natural
inclusion
Φ
bs-fld
(−)
⊆
Φ(−)
determines
a
natural
faithful
functor
C
bs-fld
→
C
which
may
be
applied
to
think
of
C
bs-fld
as
a
subcategory
of
C
[cf.
Remark
3.6.3
below].
Note
that
it
follows
immediately
from
the
existence
of
the
natural
functor
D
0
→
D
cnst
that
C
bs-fld
is
a
p-adic
Frobenioid
in
the
sense
of
[Mzk18],
Example
1.1,
(ii).
We
shall
refer
to
the
Frobenioid
C
bs-fld
obtained
in
this
way
as
the
base-field-theoretic
hull
of
the
tempered
Frobenioid
C.
Also,
we
shall
refer
to
a
morphism
of
the
Frobenioid
C
as
base-field-theoretic
if
its
zero
divisor
belongs
to
Φ
bs-fld
(−)
⊆
Φ(−).
(v)
We
shall
say
that
Φ
is
cuspidally
pure
if
the
following
conditions
are
sat-
isfied:
(a)
for
every
non-cuspidal
primary
element
x
∈
Φ(A),
where
A
∈
Ob(D),
there
exists
an
element
y
∈
Φ
bs-fld
(A)
such
that
x
≤
y;
(b)
we
have
Prime(Φ(A))
=
Prime(Φ(A))
ncsp
Prime(Φ(A))
csp
for
every
A
∈
Ob(D).
Remark
3.6.1.
Note
that
the
group-saturated-ness
hypothesis
of
Definition
3.6,
(ii),
may
be
regarded
as
the
condition
that
“divisors
relative
to
Φ
are
effective
if
and
only
if
they
are
effective
relative
to
Φ
R-log
,
i.e.,
if
and
only
if
they
are
effective
in
the
usual
sense”.
Alternatively,
this
hypothesis
[together
with
the
perf-factoriality
hypothesis
of
Definition
3.6,
(ii)]
may
be
regarded
as
the
analogue
in
the
present
“tempered
context”
of
the
monoprime-ness
hypothesis
in
[Mzk18],
Example
1.1,
(ii)
—
cf.
Remark
3.5.1.
Remark
3.6.2.
Observe
that
the
base-field-theoretic
hull
of
Definition
3.6,
(iv),
is
itself
a
tempered
Frobenioid,
and,
moreover,
that
every
p-adic
Frobenioid
may
be
obtained
in
this
way
[cf.
Remarks
3.5.1,
3.6.1].
In
particular,
it
follows
that
“the
notion
of
a
p-adic
Frobenioid
is
a
special
case
of
the
notion
of
a
tempered
Frobenioid”.
Also,
we
observe
in
passing
that
Φ
bs-fld
is
always
non-dilating
and
strictly
rational.
Remark
3.6.3.
It
follows
immediately
from
Proposition
3.4,
(ii),
and
the
ex-
plicit
divisorial
description
of
objects
and
morphisms
of
a
model
Frobenioid
given
in
[Mzk17],
Theorem
5.2,
(i)
[cf.
also
the
equivalences
of
categories
of
[Mzk17],
Definition
1.3,
(iii),
(d),
determined
by
the
operation
of
taking
the
zero
divisor
of
a
co-angular
pre-step]
that
the
objects
of
the
essential
image
[cf.
§0]
of
the
natural
functor
C
bs-fld
→
C
may
be
described
as
the
objects
of
C
that
may
be
“linked”
to
a
Frobenius-trivial
object
via
base-field
theoretic
pre-steps,
while
the
morphisms
of
the
essential
image
of
the
natural
functor
C
bs-fld
→
C
may
be
described
as
the
base-field
theoretic
morphisms
of
C
between
objects
of
the
essential
image
of
C
bs-fld
→
C.
In
particular,
the
natural
functor
C
bs-fld
→
C
is
isomorphism-full
[cf.
§0].
Thus,
no
confusion
arises
from
“identifying”
C
bs-fld
with
its
essential
image
via
the
natural
functor
C
bs-fld
→
C
in
C
[cf.
§0].
Remark
3.6.4.
If
Φ,
C
are
as
in
Definition
3.6,
(ii),
then
it
follows
from
Lemma
3.5
[applied
to
the
submonoid
Φ
⊆
Φ
R-log
]
that
the
respective
divisor
monoids
72
SHINICHI
MOCHIZUKI
Φ
pf
,
Φ
rlf
of
C
pf
,
C
rlf
also
satisfy
the
conditions
of
Definition
3.6,
(ii).
That
is
to
say,
the
perfection
and
realification
of
a
tempered
Frobenioids
are
again
tempered
Frobenioids.
Remark
3.6.5.
One
verifies
immediately
that,
when
applied
to
a
tempered
Frobenioid,
the
operations
of
perfection
and
realification
[cf.
Remark
3.6.4]
are
compatible
with
the
operation
of
passing
to
the
associated
base-field-theoretic
hull
of
the
tempered
Frobenioid.
Remark
3.6.6.
In
the
situation
of
Definition
3.6,
(ii),
if
one
supposes
further
that
Φ
is
perfect,
then
condition
(a)
follows
from
condition
(b)
[or,
alternatively,
from
the
condition
that
Φ
bs-fld
(A)
be
nonzero
for
each
A
∈
Ob(D)].
Indeed,
this
follows
immediately
by
applying
the
factorization
homomorphism
of
[Mzk17],
Definition
2.4,
(i),
(c)
[cf.
also
[Mzk17],
Definition
2.4,
(i),
(d)],
associated
to
the
perf-factorial
monoid
Φ(A).
Now
we
have
the
following
“tempered
analogue”
of
[Mzk18],
Theorem
1.2:
Theorem
3.7.
(Basic
Properties
of
Tempered
Frobenioids)
In
the
nota-
tion
of
Definition
3.6:
(i)
If
Λ
=
Z
(respectively,
Λ
=
R),
then
C
is
of
unit-profinite
(respec-
tively,
unit-trivial)
type.
For
arbitrary
Λ,
the
Frobenioid
C
is
of
isotropic,
model
[hence,
in
particular,
birationally
Frobenius-normalized],
and
sub-
quasi-Frobenius-trivial
type,
but
not
of
group-like
type.
(ii)
Suppose
D
is
of
FSMFF-type,
and
that
Φ
is
non-dilating.
Then
C
is
of
standard
type.
If,
moreover,
Φ
is
rational
[cf.
Definition
3.6,
(ii)],
then
C
is
of
rationally
standard
type.
def
(iii)
Let
A
∈
Ob(C);
A
D
=
Base(A)
∈
Ob(D).
Write
A
cnst
∈
Ob(D
cnst
)
for
the
image
of
A
D
in
D
cnst
[cf.
the
discussion
preceding
Definition
3.3].
Then
the
natural
action
of
Aut
C
(A)
on
O
(A),
O
×
(A)
factors
through
Aut
D
cnst
(A
cnst
).
If,
moreover,
Λ
∈
{Z,
Q},
then
this
factorization
determines
a
faithful
action
of
the
image
of
Aut
C
(A)
in
Aut
D
cnst
(A
cnst
)
on
O
(A),
O
×
(A).
(iv)
If
D
is
slim
[cf.
[Mzk17],
§0],
and
Λ
∈
{Z,
R},
then
C
is
also
slim.
Proof.
First,
we
consider
assertion
(i).
In
light
of
the
definition
of
C
as
a
model
Frobenioid,
it
follows
from
[Mzk17],
Theorem
5.2,
(ii),
that
C
is
of
isotropic
and
model
type;
the
fact
that
C
is
of
sub-quasi-Frobenius-trivial
type
follows
from
[Mzk17],
Proposition
1.10,
(vi).
By
Proposition
3.4,
(ii)
(respectively,
by
the
defi-
nition
of
the
realification
of
a
Frobenioid
—
cf.
[Mzk17],
Proposition
5.3),
it
follows
that
if,
moreover,
Λ
=
Z
(respectively,
Λ
=
R),
then
C
is
of
unit-profinite
(respec-
tively,
unit-trivial)
type;
the
condition
imposed
on
F
in
Definition
3.6,
(ii),
(b),
implies
immediately
that
C
is
not
of
group-like
type.
This
completes
the
proof
of
THE
ÉTALE
THETA
FUNCTION
73
assertion
(i).
As
for
assertion
(ii),
let
us
first
observe
that
since
Π
tp
X
acts
trivially
on
×
,
it
follows
[cf.
also
the
condition
imposed
on
F
in
Definition
3.6,
(ii),
(b)]
K
×
/O
K
that
every
object
of
(C
un-tr
)
birat
is
Frobenius-compact.
Thus,
assertion
(ii)
follows
immediately
from
the
definitions.
Assertion
(iii)
follows
immediately
from
Propo-
sition
3.4,
(ii).
Assertion
(iv)
follows
formally
from
[Mzk17],
Proposition
1.13,
(iii)
[since,
by
assertion
(i)
of
the
present
Theorem
3.7,
“condition
(b)”
of
loc.
cit.
is
always
satisfied
by
objects
of
C].
This
completes
the
proof
of
Theorem
3.7.
Remark
3.7.1.
We
recall
[cf.
[Mzk18],
§0]
in
passing
that
if
D
is
of
weakly
indissectible
(respectively,
strongly
dissectible;
weakly
dissectible)
type,
then
so
is
C.
Remark
3.7.2.
We
recall
in
passing
that
D
0
is
slim
[cf.
[Mzk14],
Example
3.10;
[Mzk14],
Remark
3.4.1]
and
of
FSM-,
hence
also
of
FSMFF-,
type
[cf.
[Mzk18],
Example
1.3,
(i)].
Corollary
3.8.
(Preservation
of
Base-field-theoretic
Morphisms
and
Hulls)
Suppose
that
for
i
=
1,
2,
C
i
is
a
tempered
Frobenioid
whose
base
category
D
i
is
of
FSMFF-type,
and
whose
divisor
monoid
Φ
i
is
non-dilating.
Let
∼
Ψ
:
C
1
→
C
2
be
an
equivalence
of
categories.
Then:
(i)
Suppose,
for
i
=
1,
2,
that
the
base
category
D
i
of
C
i
is
Frobenius-slim.
Then
Ψ
preserves
the
base-field-theoretic
morphisms.
(ii)
Suppose,
for
i
=
1,
2,
that
the
base
category
D
i
of
C
i
is
Div-slim
[relative
to
Φ
i
].
Then
Ψ
preserves
the
base-field-theoretic
morphisms
and
induces
a
compatible
equivalence
∼
C
1
bs-fld
→
C
2
bs-fld
of
the
subcategories
C
1
bs-fld
⊆
C
1
,
C
2
bs-fld
⊆
C
2
given
by
the
respective
base-field-
theoretic
hulls.
(iii)
Suppose
that
Ψ
preserves
the
base-field-theoretic
morphisms,
and
that
Φ
1
,
Φ
2
are
cuspidally
pure.
Then
Ψ
preserves
the
non-cuspidal
and
cuspidal
pre-steps.
If,
moreover,
Φ
1
,
Φ
2
are
rational,
then
the
induced
isomorphism
of
divisor
monoids
∼
Ψ
Φ
:
Φ
1
|
C
1
→
Φ
2
|
C
2
[lying
over
Ψ]
of
[Mzk17],
Theorem
4.9,
preserves
non-cuspidal
elements
and
primes,
as
well
as
cuspidal
elements
and
primes.
Proof.
Indeed,
by
Theorem
3.7,
(i),
(ii),
C
1
,
C
2
are
of
standard
and
isotropic
type,
but
not
of
group-like
type.
In
particular,
by
[Mzk17],
Theorem
3.4,
(ii);
[Mzk17],
Theorem
4.2,
(i),
it
follows
that
Ψ
preserves
pre-steps
and
primary
steps.
More-
over,
Ψ
is
compatible
with
the
operation
of
passing
to
the
perfection
[cf.
[Mzk17],
Theorem
3.4,
(iii)].
74
SHINICHI
MOCHIZUKI
Next,
we
consider
assertions
(i),
(ii).
By
applying
[Mzk17],
Theorem
3.4,
(iv),
in
the
case
of
assertion
(i),
and
[Mzk17],
Theorem
3.4,
(ii);
[Mzk17],
Corollary
4.11,
(ii),
in
the
case
of
assertion
(ii),
it
follows
that
Ψ
preserves
the
submonoids
“O
(−)”.
Now
observe
[cf.
Proposition
3.4,
(ii);
the
equivalences
of
categories
of
[Mzk17],
Definition
1.3,
(iii),
(d),
determined
by
the
operation
of
taking
the
zero
divisor
of
a
co-angular
pre-step]
that
a
pre-step
of
C
i
is
base-field-theoretic
if
and
only
if
its
image
A
→
B
in
C
i
pf
may
be
written
as
a
[filtered]
projective
limit
in
the
category
(C
i
pf
)
coa-pre
[where
“coa-pre”
denotes
the
subcategory
determined
B
by
the
(necessarily
co-angular)
pre-steps
of
C
i
pf
]
of
pre-steps
A
→
B
that
are
abstractly
equivalent
[cf.
§0]
to
an
endomorphism
that
belongs
to
“O
(−)”.
Thus,
Ψ
preserves
the
base-field-theoretic
pre-steps.
Note,
moreover,
that
Ψ
preserves
[cf.
[Mzk17],
Theorem
3.4,
(ii),
(iii)]
the
factorization
[cf.
[Mzk17],
Definition
1.3,
(iv),
(a)]
of
a
morphism
of
C
i
into
a
composite
of
a
morphism
of
Frobenius
type,
a
pre-step,
and
a
pull-back
morphism.
Thus,
we
conclude
that
Ψ
preserves
the
base-field-theoretic
morphisms.
This
completes
the
proof
of
assertion
(i).
Next,
to
complete
the
proof
of
assertion
(ii),
let
us
observe
that,
under
the
assumptions
of
assertion
(ii),
Ψ
preserves
[cf.
[Mzk17],
Theorem
3.4,
(iii);
[Mzk17],
Corollary
4.11,
(ii)]
the
Frobenius-trivial
objects.
Since
Ψ
preserves
base-field-theoretic
pre-
steps
and
base-field-theoretic
morphisms,
it
follows
from
the
explicit
description
of
the
base-field-theoretic
hull
given
in
Remark
3.6.3
that
Ψ
preserves
the
[objects
and
morphisms
of
the]
subcategories
C
1
bs-fld
⊆
C
1
,
C
2
bs-fld
⊆
C
2
,
hence
induces
an
∼
equivalence
of
categories
C
1
bs-fld
→
C
2
bs-fld
,
as
desired.
This
completes
the
proof
of
assertion
(ii).
Finally,
we
consider
assertion
(iii).
Since,
by
assumption,
Ψ
preserves
the
base-field-theoretic
pre-steps,
we
conclude
from
Definition
3.6,
(v),
(a)
[cf.
also
the
first
equivalence
of
categories
involving
pre-steps
of
[Mzk17],
Definition
1.3,
(iii),
(d)],
that
Ψ
preserves
the
primary
non-cuspidal
steps,
hence,
[by
Definition
3.6,
(v),
(b)]
that
Ψ
preserves
the
primary
cuspidal
steps.
Thus,
by
considering
the
“factorization
homomorphisms”
arising
from
the
fact
that
Φ
1
,
Φ
2
are
perf-factorial
[cf.
[Mzk17],
Definition
2.4,
(i),
(c)]
in
the
context
of
the
perfections
of
C
1
,
C
2
,
it
follows
that
Ψ
preserves
the
non-cuspidal
and
cuspidal
pre-steps.
The
remainder
of
∼
assertion
(iii)
now
follows
immediately
from
the
isomorphism
Ψ
Φ
:
Φ
1
|
C
1
→
Φ
2
|
C
2
of
[Mzk17],
Theorem
4.9
[which
is
applicable
in
light
of
Theorem
3.7,
(ii)].
This
completes
the
proof
of
assertion
(iii).
Remark
3.8.1.
Note
that
in
the
situation
of
Corollary
3.8,
(ii),
for
suitable
base
categories
[i.e.,
of
the
sort
that
appear
in
[Mzk18],
Theorem
2.4]
one
may
apply
∼
to
the
equivalence
of
categories
C
1
bs-fld
→
C
2
bs-fld
induced
by
Ψ
the
theory
of
the
category-theoreticity
of
the
Kummer
and
reciprocity
maps,
as
discussed
in
[Mzk18],
Theorem
2.4.
Remark
3.8.2.
In
the
situation
of
Corollary
3.8,
suppose
further
that
Ψ
preserves
the
base-field-theoretic
morphisms,
and
that
Φ
1
,
Φ
2
are
cuspidally
pure
and
rational
[cf.
Corollary
3.8,
(iii)].
Then
observe
that
by
considering
zero
divisors
of
base-
field-theoretic
pre-steps
as
in
the
proof
of
Corollary
3.8,
(i),
(ii),
it
follows
that
[in
THE
ÉTALE
THETA
FUNCTION
75
def
the
notation
of
Corollary
3.8],
for
C
1
∈
Ob(C
1
),
C
2
=
Ψ(C
1
),
non-cuspidal
primes
p
1
,
q
1
of
Φ(C
1
)
such
that
p
1
→
p
2
,
q
1
→
q
2
[where
p
2
,
q
2
∈
Prime(Φ(C
2
))],
we
obtain,
for
i
=
1,
2,
natural
isomorphisms
∼
rlf
(R
≥0
∼
=)
Φ
i
(C
i
)
rlf
p
i
→
Φ
i
(C
i
)
q
i
(
∼
=
R
≥0
)
[i.e.,
induced
by
considering
the
zero
divisors
of
elements
of
O
(C
i
)]
which
are
∼
compatible
with
the
isomorphism
Φ
1
(C
1
)
→
Φ
2
(C
2
)
induced
by
Ψ
Φ
.
Finally,
we
begin
to
relate
the
theory
of
tempered
Frobenioids
to
the
theory
of
the
étale
theta
function,
as
discussed
in
§1,
§2:
Example
3.9.
Theta
Functions
and
Tempered
Frobenioids.
(i)
Suppose
that
X
log
is
a
smooth
log
orbicurve
of
the
sort
defined
in
Definition
2.5,
(i),
(ii)
[i.e.,
one
of
the
following
smooth
log
orbicurves:
“X
log
”,
“C
log
”,
“X
log
”,
log
log
log
log
“C
log
”,
“
Ẋ
”,
“
Ċ
”,
“
Ẋ
”,
“
Ċ
”].
Then
there
exists
a
[1-]commutative
dia-
gram
of
finite
log
étale
Galois
coverings
of
smooth
log
orbicurves
U
log
⏐
⏐
→
X
log
⏐
⏐
Y
log
→
W
log
—
where
U
log
,
Y
log
are
smooth
log
curves
that
arise
as
generic
fibers
of
stable
log
curves
U
log
,
Y
log
over
[formal
spectra
equipped
with
appropriate
log
structures
determined
by]
rings
of
integers
of
appropriate
finite
extensions
of
K;
the
diagram
∼
induces
a
natural
isomorphism
Gal(U
log
/X
log
)
→
Gal(Y
log
/W
log
);
the
order
of
the
group
Gal(U
log
/X
log
)
∼
=
Gal(Y
log
/W
log
)
is
≤
2;
Y
log
→
W
log
is
unramified
at
the
cusps
of
Y
log
;
Y
log
is
of
genus
1.
[Thus,
for
instance,
when
X
log
is
“
Ċ
take
the
upper
arrow
of
the
diagram
to
be
“
Ẋ
the
diagram
to
be
“
Ẋ
log
→
Ċ
def
log
log
→
Ċ
log
log
”,
one
may
”
and
the
lower
arrow
of
”.]
Write
D
U
=
B
temp
(U
log
)
0
;
def
D
X
=
B
temp
(X
log
)
0
(=
D
0
)
def
D
Y
=
B
temp
(Y
log
)
0
;
def
D
W
=
B
temp
(W
log
)
0
—
so
the
above
[1-]commutative
diagram
induces
natural
functors
D
U
→
D
X
,
D
U
→
D
Y
,
D
X
→
D
W
,
D
Y
→
D
W
[obtained
by
regarding
a
tempered
covering
of
the
domain
orbicurve
of
an
arrow
of
the
above
commutative
diagram
as
a
tempered
covering
of
the
codomain
curve
of
the
arrow].
(ii)
Let
us
write
D
Y
ell
⊆
D
Y
;
ell
D
W
⊆
D
W
for
the
full
subcategories
of
tempered
coverings
that
are
unramified
over
the
cusps
of
Y
log
,
W
log
[i.e.,
the
“tempered
coverings
of
the
underlying
elliptic
curve
of
Y
log
”].
76
SHINICHI
MOCHIZUKI
Thus,
by
taking
the
left
adjoints
to
the
natural
inclusion
functors
D
Y
ell
→
D
Y
,
ell
ell
→
D
W
,
we
obtain
natural
functors
D
Y
→
D
Y
ell
,
D
W
→
D
W
[cf.
[Mzk18],
D
W
Example
1.3,
(ii)],
as
well
as
1-commutative
diagrams
of
natural
functors
D
Y
ell
⏐
⏐
→
D
Y
⏐
⏐
D
Y
⏐
⏐
→
D
Y
ell
⏐
⏐
ell
D
W
→
D
W
D
W
→
ell
D
W
[since
Y
log
→
W
log
is
unramified
at
the
cusps
of
Y
log
].
(iii)
Next,
let
us
denote
by
Φ
W
the
monoid
on
D
W
given
by
forming
the
perfection
of
the
monoid
“Φ
0
”
of
Definition
3.3,
(iii),
for
some
choice
of
tempered
filter
on
W
log
that
arises
from
a
tempered
filter
on
Y
log
[i.e.,
whose
constituent
tp
tp
subgroups
⊆
Δ
tp
W
are
contained
in
Δ
Y
⊆
Δ
W
].
Now
define
ell
Φ
ell
W
ell
⊆
Φ
W
|
D
W
ell
as
follows:
For
A
∈
Ob(D
W
),
we
take
Φ
ell
(A)
to
be
the
perf-saturation
[cf.
§0]
in
W
ell
Φ
W
(A)
of
the
submonoid
log
Gal(Z
∞
/A)
lim
Div
+
(Z
log
⊆
Φ
W
(A)
∞
)
−→
log
Z
∞
log
log
ell
—
where
Z
∞
ranges
over
the
connected
tempered
coverings
Z
∞
→
A
in
D
W
such
log
log
that
the
composite
covering
Z
∞
→
A
→
W
arises
as
the
generic
fiber
of
the
“universal
combinatorial
covering”
Z
log
of
the
stable
logarithmic
model
Z
log
of
some
∞
ell
finite
log
étale
Galois
covering
Z
log
→
W
log
[in
D
W
!]
with
stable,
split
reduction
over
the
ring
of
integers
of
a
finite
extension
L
of
K;
the
superscript
Galois
group
denotes
the
submonoid
of
elements
fixed
by
the
Galois
group
in
question.
Here,
we
pause
to
observe
that
the
various
monoids
that
occur
in
the
above
inductive
limit
are
all
contained
in
Φ
W
(A),
and
that
the
induced
morphisms
on
perf-saturations
be-
tween
these
monoids
are
bijective
[cf.
Remark
3.3.1].
Indeed,
this
bijectivity
follows
immediately
from
the
well-known
structure
of
the
special
fibers
of
the
“universal
combinatorial
coverings”
that
appear
in
the
above
inductive
limit
[i.e.,
“chains
of
copies
of
the
projective
line”
—
cf.,
e.g.,
the
discussion
preceding
Proposition
1.1].
Set
def
ell
ell
)|
D
⊆
Φ
W
Φ
ell
W
=
Φ
W
ell
|
D
W
⊆
(Φ
W
|
D
W
W
ell
—
where
“|
D
W
”
is
with
respect
to
the
functor
D
W
→
D
W
defined
in
(ii).
Now
ell
ell
observe
that
Φ
W
is
a
perfect
[cf.
the
definition
of
Φ
W
ell
as
a
perf-saturation
inside
the
ell
ell
on
D
perfect
monoid
Φ
W
|
D
W
W
]
and
[manifestly
—
cf.
Remark
3.6.1]
group-saturated
submonoid
of
the
monoid
Φ
W
on
D
W
,
which
is,
moreover,
perf-factorial,
non-
dilating
[cf.
Proposition
3.4,
(i);
the
above
observation
concerning
the
bijectivity
of
induced
morphisms
on
perf-saturations],
and
cuspidally
pure
[cf.
the
well-known
structure
of
the
special
fibers
of
the
“universal
combinatorial
coverings”
that
appear
in
the
above
inductive
limit].
Also,
Φ
ell
W
is
[manifestly]
independent,
up
to
natural
isomorphism,
of
the
choice
of
tempered
filter
on
W
used
to
define
Φ
W
.
THE
ÉTALE
THETA
FUNCTION
77
(iv)
If
α
:
A
→
B
is
any
morphism
of
D
W
,
then
set
def
D
α
=
(D
W
)
B
[α]
(⊆
(D
W
)
B
)
—
where
we
regard
α
as
an
object
of
(D
W
)
B
–
cf.
the
notational
conventions
of
§0;
[Mzk17],
§0.
Thus,
D
α
is
a
quasi-temperoid
[cf.
[Mzk14],
Definition
A.1,
(ii)].
Also,
we
observe
that
D
W
,
D
X
,
D
Y
,
D
U
are
special
cases
of
“D
α
”
[obtained
by
taking
“α”
be
the
identity
morphism
of
W
log
,
X
log
,
Y
log
,
U
log
].
Note
that
we
have
a
natural
functor
D
α
→
D
W
.
Now
let
us
write
def
ell
Φ
ell
α
=
Φ
W
|
D
α
(⊆
Φ
W
|
D
α
)
and
note
that
it
follows
immediately
from
the
above
discussion
that
this
[sub]monoid
ell
Φ
ell
α
—
which
is
obtained
simply
by
restricting
the
functor
Φ
W
via
some
functor
—
is
perfect,
group-saturated,
perf-factorial,
non-dilating,
and
cuspidally
pure.
Note,
moreover,
that
the
existence
of
the
theta
functions
discussed
in
§1
[cf.
especially
the
description
of
the
zeroes
and
poles
of
these
theta
functions
given
in
Proposition
1.4,
(i)]
implies
that
the
monoid
Φ
ell
α
is
also
rational
[cf.
Definition
3.6,
(ii);
[Mzk17],
Definition
4.5,
(ii)].
In
particular,
it
follows
that:
This
monoid
Φ
ell
α
[along
with
its
perfection
and
realification
—
cf.
Remark
3.6.4]
gives
rise
to
a
tempered
Frobenioid
[cf.
Definition
3.6,
(ii);
Remark
3.6.6]
of
rationally
standard
type
[cf.
Theorem
3.7,
(ii)]
with
perfect
divisor
monoid
over
the
slim
[cf.
Remark
3.7.2]
base
category
D
α
of
FSM-type
[cf.
Remark
3.7.2].
78
SHINICHI
MOCHIZUKI
Section
4:
General
Bi-Kummer
Theory
In
the
present
§4,
we
apply
the
theory
of
tempered
Frobenioids
developed
in
§3
to
discuss
the
analogue,
for
log-meromorphic
functions
on
tempered
coverings
of
smooth
log
orbicurves
over
nonarchimedean
[mixed-characteristic]
local
fields,
of
the
Kummer
theory
for
p-adic
Frobenioids
developed
in
[Mzk18],
§2.
One
im-
portant
aspect
[i.e.,
in
a
word,
the
“bi”
portion
of
the
term
“bi-Kummer”]
of
the
“bi-Kummer
theory”
theory
developed
here
—
by
comparison,
for
instance,
to
the
Kummer
theory
for
arbitrary
Frobenioids
discussed
in
[Mzk18],
Definition
2.1
—
is
that
instead
of
just
taking
roots
of
the
given
log-meromorphic
function,
one
con-
siders
roots
of
the
pair
of
sections
of
a
line
bundle
that
correspond,
respectively,
to
the
“numerator”
and
“denominator”
of
the
log-meromorphic
function
[cf.
Re-
mark
4.3.1
below].
Another
important
feature
of
the
theory
developed
here
—
by
comparison
to
the
theory
of
[Mzk18],
§2,
for
p-adic
Frobenioids
—
is
the
absence
of
an
analogue
of
the
reciprocity
map
[cf.
Remark
4.4.1
below].
As
we
shall
see
in
§5
below
[cf.
Theorem
5.6
and
its
proof],
the
additional
“layer
of
complexity”
that
arises
from
the
former
feature
has
the
effect
of
compensating
[to
a
certain
extent,
at
least
in
the
case
of
the
situation
discussed
in
§2]
for
the
“handicap”
constituted
by
the
latter
feature.
Finally,
we
remark
that
the
theory
developed
here
may
be
regarded
as
and,
indeed,
was
motivated
by
the
goal
of
developing
a
Frobenioid-
theoretic
translation/generalization
—
via
the
theory
of
base-Frobenius
pairs
[cf.
[Mzk17],
Definition
2.7,
(iii);
[Mzk17],
Proposition
5.6]
—
of
the
scheme-theoretic
constructions
of
§1.
Let
X
log
,
K,
D
0
=
B
temp
(X
log
)
0
be
as
in
§3.
In
the
following
discussion,
we
fix
a
tempered
Frobenioid
C
whose
monoid
type
is
Z,
whose
divisor
monoid
Φ
is
perfect,
whose
base
category
D
is
of
the
form
def
D
=
D
0
[D]
(⊆
D
0
)
[cf.
§0],
where
D
∈
Ob(D
0
),
and
whose
base-field-theoretic
hull
we
denote
by
def
C
bs-fld
⊆
C.
Also,
we
fix
a
Frobenius-trivial
object
A
∈
Ob(C)
such
that
A
bs
=
Base(A
)
∈
Ob(D)
is
a
Galois
[cf.
[Mzk14],
Definition
3.1,
(iv)]
object.
Thus,
A
bs
determines
normal
open
subgroups
H
⊆
Π
tp
X
;
bs-fld
H
⊆
G
K
bs-fld
[i.e.,
H
is
the
image
of
H
in
G
K
]
of
the
tempered
fundamental
group
Π
tp
X
of
tp
log
and
the
quotient
Π
X
G
K
determined
by
the
absolute
Galois
group
of
K.
X
In
the
following
discussion,
we
shall
use
the
superscript
“birat”
(respectively,
“bs”)
to
denote
the
object
or
arrow
determined
by
a
given
object
or
arrow
of
C
in
the
birationalization
C
birat
[cf.
[Mzk17],
Proposition
4.4]
(respectively,
base
category
D)
of
C.
Recall
that
pre-steps
of
C
map
to
isomorphisms
in
C
birat
[cf.
[Mzk17],
Propo-
sition
4.4,
(iv)].
In
particular,
it
follows
that
any
base-equivalent
pair
of
pre-steps
THE
ÉTALE
THETA
FUNCTION
79
s
,
s
:
A
→
B
in
C
determines,
by
inverting
the
image
of
s
in
C
birat
,
an
element
“s
·
(s
)
−1
”
∈
O
×
(A
birat
).
Definition
4.1.
Let
A
∈
Ob(C).
(i)
If
f
∈
O
×
(A
birat
),
then
we
shall
refer
to
as
a
fraction-pair,
or,
alternatively,
as
a
right
fraction-pair
[for
f
],
any
base-equivalent
pair
of
pre-steps
s
,s
:
A
→
B
such
that
s
·
(s
)
−1
=
f
∈
O
×
(A
birat
),
and,
moreover,
Div(s
),
Div(s
)
have
disjoint
supports
[cf.
[Mzk17],
Proposition
4.1,
(iii)].
[Thus,
Div(s
),
Div(s
)
are
uniquely
determined
by
f
.]
In
this
situation,
we
shall
refer
to
Div(s
)
as
the
zero
divisor
and
to
Div(s
)
as
the
divisor
of
poles
of
the
fraction-pair
(s
,
s
);
we
shall
refer
to
A
(respectively,
B)
as
the
domain
(respectively,
codomain)
of
the
fraction-
pair
(s
,
s
);
if
we
denote
by
f
|
B
∈
O
×
(B
birat
)
the
element
determined
by
(s
,
s
)
[cf.
[Mzk17],
Proposition
4.4,
(iv)],
then
we
shall
refer
to
the
pair
(s
,
s
)
as
a
left
fraction-pair
[for
f
|
B
].
(ii)
We
shall
say
that
A
is
Galois
if
A
bs
∈
Ob(D)
is
Galois.
Suppose
that
A
is
Galois.
Then,
by
the
definition
of
D,
there
is
a
natural
surjective
outer
homomor-
phism
bs
Π
tp
X
Aut
D
(A
)
[cf.
the
discussion
of
[Mzk18],
Definition
2.2,
(i),
in
the
case
of
p-adic
Frobenioids];
write
H
A
bs
⊆
Aut
D
(A
bs
)
for
the
image
of
H
via
this
surjection
[which
is
well-defined,
since
H
is
normal]
and
H
A
⊆
Aut
C
(A)/O
×
(A)
for
the
inverse
image
of
H
A
bs
via
the
natural
injection
Aut
C
(A)/O
×
(A)
→
Aut
D
(A
bs
).
If
the
natural
injection
H
A
→
H
A
bs
is
a
bijection,
then
we
shall
say
that
A
is
H
-
ample.
(iii)
Suppose
that
A
is
H
-ample
[hence,
in
particular,
Galois],
and
that
f
∈
O
×
(A
birat
)
is
an
element
fixed
by
the
natural
action
of
H
A
;
let
N
∈
N
≥1
.
Then
we
shall
say
that
A
is
(N,
H
,
f
)-saturated
if
the
following
conditions
are
satisfied:
(a)
there
exist
pre-steps
A
→
A,
A
→
A
in
C,
where
A
is
Frobenius-trivial
[hence
determines
an
object
of
the
p-adic
Frobenioid
C
bs-fld
—
cf.
Remark
3.6.3],
such
that
bs-fld
A
is
(N,
H
)-saturated
[cf.
[Mzk18],
Definition
2.2,
(ii)]
as
an
object
of
C
bs-fld
;
(b)
there
exists
a
g
∈
O
×
(A
birat
)
such
that
g
N
=
f
.
(iv)
We
shall
say
that
a
morphism
α
:
A
→
B
of
C
is
of
base-Frobenius
type
if
there
exist
a
subgroup
G
⊆
Aut
C
B
(α)
⊆
Aut
C
(A)
and
a
factorization
α
=
α
◦
α
such
that
the
following
conditions
are
satisfied:
(a)
A
is
Frobenius-trivial,
Galois,
def
and
μ
N
-saturated
[cf.
[Mzk18],
Definition
2.1,
(i)],
where
N
=
deg
Fr
(α);
(b)
G
maps
isomorphically
to
Gal(A
bs
/B
bs
)
⊆
Aut
D
(A
bs
);
(c)
α
is
a
base-identity
80
SHINICHI
MOCHIZUKI
endomorphism
of
Frobenius
type;
(d)
α
is
a
pull-back
morphism;
(e)
G,
α
,
α
arise
from
a
base-Frobenius
pair
of
C
[cf.
Theorem
3.7,
(i);
[Mzk17],
Proposition
5.6].
In
this
situation,
we
shall
refer
to
the
subgroup
G
and
the
factorization
α
=
α
◦
α
as
being
of
base-Frobenius
type.
Remark
4.1.1.
Note
that
in
the
situation
of
Definition
4.1,
(iv),
if
α
:
A
→
B
is
of
base-Frobenius
type,
then
by
applying
Proposition
3.4,
(ii),
together
with
the
factorization
of
[Mzk17],
Definition
1.3,
(iv),
(a),
one
verifies
easily
that
A
→
B
is
a
categorical
quotient
[cf.
[Mzk17],
§0]
of
A
by
the
subgroup
G
·
μ
N
(A)
⊆
Aut
C
(A)
in
the
full
subcategory
of
C
determined
by
the
Frobenius-trivial
objects
[cf.
[Mzk17],
Theorem
5.1,
(iii)].
Proposition
4.2.
(Construction
of
Bi-Kummer
Data
I:
Roots
of
Fraction-
);
s
,
s
:
A
→
Pairs)
In
the
notation
of
the
above
discussion,
let
f
∈
O
×
(A
birat
B
a
right
fraction-pair
for
f
[i.e.,
a
left
fraction-pair
for
f
|
B
];
N
∈
N
≥1
.
Then:
(i)
A
pair
of
morphisms
t
,
t
:
A
→
C
is
a
right
fraction-pair
for
f
if
∼
and
only
if
there
exists
a
[necessarily
unique]
isomorphism
v
:
B
→
C
such
that
t
=
v
◦
s
,
t
=
v
◦
s
.
(ii)
A
pair
of
morphisms
t
,
t
:
C
→
B
is
a
left
fraction-pair
for
f
|
B
if
∼
and
only
if
there
exists
a
[necessarily
unique]
isomorphism
v
:
C
→
A
such
that
t
=
s
◦
v,
t
=
s
◦
v.
(iii)
There
exist
commutative
diagrams
in
C
s
N
A
N
−→
⏐
⏐
α
A
s
−→
s
B
N
⏐
⏐
β
N
A
N
−→
⏐
⏐
α
B
A
s
−→
B
N
⏐
⏐
β
B
—
where
α,
β
are
isometries
of
Frobenius
degree
N
;
α
is
of
base-Frobenius
def
type,
with
factorization
of
base-Frobenius
type
α
=
α
◦
α
;
f
|
A
N
=
((α
)
birat
)
∗
(f
)
[cf.
[Mzk17],
Proposition
1.11,
(iv)];
A
N
is
(N,
H
,
f
|
A
N
)-saturated;
s
N
,
s
N
:
A
N
→
B
N
are
base-equivalent
pre-steps.
In
particular,
there
exists
an
element
birat
)
(respectively,
∈
O
×
(B
N
))
for
which
(s
N
,
s
N
)
is
a
right
(respec-
∈
O
×
(A
birat
N
tively,
left)
fraction-pair,
and
whose
N
-th
power
is
equal
to
f
|
A
N
(respectively,
def
f
|
B
N
=
(f
|
A
N
)|
B
N
[cf.
the
notation
of
Definition
4.1,
(i)]).
In
the
following,
we
shall
refer
to
such
a
pair
of
commutative
diagrams
as
an
N
-th
root
of
the
fraction-pair
(s
,
s
),
to
A
N
as
the
N
-domain
of
this
root
of
a
fraction-pair,
and
to
B
N
as
the
N
-codomain
of
this
root
of
a
fraction-pair.
(iv)
We
continue
to
use
the
notation
of
(iii).
Let
s
N
A
N
−→
⏐
⏐
α
A
s
−→
s
B
N
⏐
⏐
β
N
A
N
−→
⏐
⏐
α
B
A
s
−→
B
N
⏐
⏐
β
B
THE
ÉTALE
THETA
FUNCTION
81
be
another
N
-th
root
of
a
left
fraction-pair
(s
,
s
)
for
f
|
B
;
δ
∈
Aut
C
(A
)
the
∼
bs
unique
automorphism
[cf.
(ii)]
such
that
s
=
s
◦
δ,
s
=
s
◦
δ;
A
:
A
bs
N
→
A
N
an
isomorphism
of
D
such
that
α
bs
=
δ
bs
◦
α
bs
◦
A
.
Then,
after
possibly
replacing
s
N
by
u
◦
s
N
,
for
some
u
∈
μ
N
(B
N
)
[where
“μ
N
(−)”
is
as
in
[Mzk18],
Definition
∼
∼
2.1,
(i)],
there
exist
isomorphisms
ζ
A
:
A
N
→
A
N
,
ζ
B
:
B
N
→
B
N
in
C
which
fit
into
commutative
diagrams
s
N
A
N
−→
⏐
⏐
ζ
A
A
N
s
N
−→
s
B
N
⏐
⏐
ζ
B
N
A
N
−→
⏐
⏐
ζ
A
B
N
A
N
s
N
−→
B
N
⏐
⏐
ζ
B
B
N
bs
and,
moreover,
satisfy
α
=
δ
◦
α
◦
ζ
A
,
β
=
β
◦
ζ
B
,
ζ
A
=
A
.
Here,
ζ
B
is
uniquely
determined
by
ζ
A
;
ζ
A
is
uniquely
determined
by
A
,
up
to
composition
with
an
element
of
μ
N
(A
N
).
[Thus,
ζ
B
is
uniquely
determined
by
A
,
up
to
composition
with
an
element
of
μ
N
(B
N
).]
In
the
following,
we
shall
refer
to
such
a
pair
(ζ
A
,
ζ
B
)
as
an
isomorphism
between
the
two
given
N
-th
roots
of
fraction-pairs.
Proof.
The
sufficiency
portion
of
assertion
(i)
is
immediate.
To
verify
the
necessity
portion
of
assertion
(i),
observe
that
by
the
equivalences
of
categories
of
[Mzk17],
Definition
1.3,
(iii),
(d),
the
“disjoint
supports”
condition
[cf.
Definition
4.1,
(i)]
on
the
Div(−)’s
of
the
components
of
a
fraction-pair
t
,
t
:
A
→
C
implies
the
∼
existence
of
isomorphisms
v,
v
:
B
→
C
such
that
t
=
v
◦
s
,
t
=
v
◦
s
;
since,
moreover,
t
,
t
are
base-equivalent,
it
follows
that
v
=
v
◦
v
,
for
some
v
∈
O
×
(C
).
On
the
other
hand,
since
the
fraction-pair
t
,
t
:
A
→
C
determines
the
same
element
of
O
×
(A
birat
)
as
the
fraction-pair
s
,
s
:
A
→
C
,
we
thus
conclude
that
v
=
1.
Note
that
the
uniqueness
of
v
follows
from
the
total
epimorphicity
of
C.
This
completes
the
proof
of
assertion
(i).
The
proof
of
assertion
(ii)
is
entirely
similar
[except
that
one
concludes
uniqueness
from
the
fact
that
pre-steps
are
always
monomorphisms
—
cf.
[Mzk17],
Definition
1.3,
(v),
(a)].
Next,
we
consider
assertions
(iii),
(iv).
First,
let
us
observe
that,
by
the
def-
inition
of
“log-meromorphic”
[cf.
Definition
3.1,
(ii)],
it
follows
immediately
that
over
some
tempered
covering
of
X
log
that
occurs
as
the
“universal
combinatorial
covering”
of
a
finite
étale
covering
of
X
log
with
stable,
split
reduction,
f
admits
an
N
-th
root.
Since,
moreover,
the
divisor
monoid
Φ
is
assumed
to
be
perfect,
it
follows
that
the
divisors
of
zeroes
and
poles
of
such
an
N
-th
root
belong
to
Φ(−)
of
the
tempered
covering
in
question.
Now
since
the
Frobenioid
C
is
of
model
[hence,
in
particular,
pre-model
—
cf.
[Mzk17],
Definition
4.5,
(i)]
type
[cf.
Theorem
3.7,
(i)],
it
follows
that
C
admits
a
base-Frobenius
pair
[cf.
[Mzk17],
Definition
2.7,
(iii)].
Thus,
the
existence
of
a
pair
of
commutative
diagrams
as
in
the
statement
of
asser-
tion
(iii)
follows
by
translating
the
above
“scheme-theoretic
observations”
into
the
language
of
Frobenioids
—
cf.
[Mzk17],
Definition
1.3,
(iii),
(d)
[on
the
existence
of
pre-steps
with
prescribed
zero
divisor];
[Mzk18],
Remark
2.2.1
[concerning
the
issue
bs-fld
)-saturation”].
of
“(N,
H
Finally,
to
verify
the
uniqueness
up
to
isomorphism
of
such
a
pair
of
commu-
tative
diagrams
as
stated
in
assertion
(iv),
let
us
first
observe
that
by
replacing
82
SHINICHI
MOCHIZUKI
α,
s
,
s
by
δ
−1
◦
α,
s
◦
δ,
s
◦
δ,
respectively,
we
may
assume
without
loss
of
generality
that
δ
is
the
identity.
Next,
let
us
observe
that
it
follows
from
the
“(N,
H
,
f
|
A
N
)-saturated-ness”
condition
in
the
statement
of
assertion
(iii)
[cf.
also
Proposition
3.4,
(ii)]
that
the
pull-back
∈
O
×
(A
N
)
or
∈
O
×
(A
N
)
of
any
element
∈
O
×
(A
)
[i.e.,
via
the
“pull-back
portion”
of
α,
α
—
cf.
Definition
4.1,
(iv),
(d)]
admits
an
N
-th
root.
Now
[in
light
of
this
observation]
the
existence
of
a
ζ
A
as
desired
follows
immediately
from
the
uniqueness
up
to
conjugation
by
a
unit
of
base-Frobenius
pairs
of
A
N
,
A
N
[cf.
[Mzk17],
Proposition
5.6],
by
thinking
of
α,
α
as
categorical
quotients
[cf.
Remark
4.1.1]
and
applying
the
base-triviality
and
Aut-
ampleness
of
the
full
subcategory
of
C
determined
by
the
Frobenius-trivial
objects
[cf.
[Mzk17],
Theorem
5.1,
(iii)].
The
existence
of
a
ζ
B
as
desired
follows
from
the
equivalences
of
categories
determined
by
pre-steps
of
[Mzk17],
Definition
1.3,
(iii),
(d).
The
essential
uniqueness
of
ζ
A
,
ζ
B
as
asserted
follows
immediately
from
the
various
conditions
imposed
on
ζ
A
,
ζ
B
.
This
completes
the
proof
of
assertion
(iv).
Proposition
4.3.
(Construction
of
Bi-Kummer
Data
II:
Bi-Kummer
Roots)
In
the
notation
of
Proposition
4.2,
(iii):
(i)
Let
s
trv
N
:
H
A
N
→
Aut
C
(A
N
)
be
the
group
homomorphism
arising
from
a
base-Frobenius
pair
of
A
N
[cf.
The-
orem
3.7,
(i);
[Mzk17],
Proposition
5.6].
[Thus,
s
trv
N
is
completely
determined
up
to
conjugation
by
an
element
of
O
×
(A
N
)
—
cf.
Theorem
3.7,
(i);
[Mzk17],
Propo-
sition
5.6.]
Then
there
exist
unique
group
homomorphisms
s
N
-gp
:
H
B
N
→
Aut
C
(B
N
);
s
N
-gp
:
H
B
N
→
Aut
C
(B
N
)
∼
such
that,
relative
to
the
isomorphism
H
A
N
→
H
B
N
determined
by
the
[base-
equivalent!]
pair
of
morphisms
s
N
,
s
N
,
we
have
s
N
-gp
(h)
◦
s
N
=
s
N
◦
(s
trv
N
|
H
BN
)(h);
s
N
-gp
(h)
◦
s
N
=
s
N
◦
(s
trv
N
|
H
BN
)(h)
for
all
h
∈
H
B
N
.
[In
particular,
it
follows
that
B
N
is
H
-ample.]
In
the
following,
we
shall
refer
to
such
a
pair
(s
N
-gp
,
s
N
-gp
)
as
a
bi-Kummer
N
-th
root
of
the
fraction-pair
(s
,
s
);
also,
we
shall
speak
of
A
N
,
B
N
as
being
the
“N
-domain”,
“N
-codomain”,
respectively,
[cf.
Proposition
4.2,
(iii)]
not
only
of
the
original
root
of
a
fraction-pair,
also
of
the
resulting
bi-Kummer
root.
(ii)
In
the
notation
of
(i),
the
collection
of
bi-Kummer
N
-th
roots,
with
N
-
codomain
B
N
,
of
a
fraction-pair
with
domain
isomorphic
to
A
,
of
some
rational
function
whose
pull-back
to
[the
N
-codomain]
B
N
is
equal
to
f
|
B
N
,
is
equal
to
the
collection
of
pairs
obtained
from
(s
N
-gp
,
s
N
-gp
)
by
(a)
simultaneous
conjugation
by
an
element
ζ
Aut
∈
O
×
(B
N
),
followed
by
THE
ÉTALE
THETA
FUNCTION
83
(b)
non-simultaneous
conjugation
[of,
say,
s
N
-gp
,
but
not
s
N
-gp
]
by
an
element
u
∈
μ
N
(B
N
)
[cf.
the
“u”
of
Proposition
4.2,
(iv)].
(iii)
In
the
notation
of
(i),
the
difference
s
N
-gp
·
(s
N
-gp
)
−1
determines
a
twisted
homomorphism
H
B
N
→
μ
N
(B
N
),
hence
an
element
of
the
cohomology
module
H
1
(H
B
N
,
μ
N
(B
N
)),
which
is
equal
to
the
Kummer
class
[cf.
[Mzk18],
Definition
2.1,
(ii)]
κ
f
|
BN
∈
H
1
(H
B
N
,
μ
N
(B
N
))
of
f
|
B
N
[cf.
the
notation
of
Proposition
4.2,
(iii)].
In
particular,
this
cohomol-
ogy
class
is
independent
of
the
[simultaneous
and
non-simultaneous]
conjugation
operations
discussed
in
(ii).
Proof.
First,
we
consider
assertion
(i).
To
prove
the
existence
and
uniqueness
of
s
N
-gp
,
s
N
-gp
,
it
follows
from
the
general
theory
of
Frobenioids
—
cf.
the
first
equivalence
of
categories
involving
pre-steps
of
[Mzk17],
Definition
1.3,
(iii),
(d);
the
fact
that
Frobenioids
are
always
totally
epimorphic
—
that
it
suffices
to
prove
that
Div(s
N
),
Div(s
N
)
∈
Φ(A
N
)
are
fixed
by
H
A
N
.
But
since
N
·
Div(s
N
),
N
·
Div(s
N
)
∈
Φ(A
N
)
arise
as
pull-backs
to
A
N
of
elements
of
Φ(A
)
[i.e.
Div(s
),
Div(s
)],
this
follows
from
the
fact
that
[by
definition]
H
acts
trivially
on
A
bs
[together
with
the
fact
the
monoid
Φ(A
N
)
is
torsion-free!].
This
completes
the
proof
of
assertion
(i).
Assertion
(iii)
follows
immediately
from
the
definitions
[cf.,
especially,
[Mzk18],
Definition
2.1,
(ii)].
Finally,
we
observe
that
assertion
(ii)
follows
immediately
by
applying
Propo-
sition
4.2,
(ii),
to
the
fraction-pairs
(s
N
,
s
N
)
and
(s
N
,
s
N
)
of
Proposition
4.2,
(iv)
[after
possibly
replacing
s
N
by
u
◦
s
N
].
Indeed,
since
s
trv
N
is
completely
determined
up
to
conjugation
by
an
element
of
O
×
(A
N
)
[cf.
Theorem
3.7,
(i);
[Mzk17],
Propo-
sition
5.6.],
we
thus
conclude
that
the
collection
of
bi-Kummer
N
-th
roots
under
consideration
is
as
stated
in
assertion
(ii).
This
completes
the
proof
of
assertion
(ii).
Remark
4.3.1.
Of
course,
even
without
applying
the
somewhat
nontrivial
the-
ory
of
base-Frobenius
pairs
[i.e.,
[Mzk17],
Proposition
5.6],
liftings
to
Aut
C
(A
N
)
of
individual
elements
of
H
A
N
[cf.
the
notation
of
Proposition
4.3,
(i)]
are
completely
determined
up
to
possible
translation
by
elements
of
O
×
(A
N
).
The
crucial
differ-
ence,
however,
between
an
indeterminacy
up
to
translation
by
elements
of
O
×
(A
N
)
and
an
indeterminacy
up
to
conjugation
by
elements
of
O
×
(A
N
)
is
that,
unlike
the
-gp
,
s
N
-gp
]
which
former,
the
latter
allows
one
to
work
with
sections
[i.e.,
s
trv
N
,
s
N
are
group
homomorphisms.
Of
course,
the
Kummer
class
[which,
by
Proposition
4.3,
(iii),
is
determined
by
the
pair
(s
N
-gp
,
s
N
-gp
)]
is
always
a
[twisted]
homomor-
phism.
On
the
other
hand,
the
theory
of
base-Frobenius
pairs
allows
one
to
work
with
group
homomorphisms
[i.e.,
s
N
-gp
or
s
N
-gp
]
even
when
one
is
forced
to
restrict
one’s
attention
to
only
one
of
the
two
arrows
s
N
or
s
N
,
i.e.,
in
situations
where
one
is
not
allowed
to
work
with
the
fraction-pair
as
a
“single
entity”
—
cf.
the
84
SHINICHI
MOCHIZUKI
theory
of
“Frobenioid-theoretic
mono-theta
environments”
developed
in
Theorem
5.10
below.
Remark
4.3.2.
Note
that
if
N
divides
N
∈
N
≥1
,
then
one
verifies
immediately
that
given
an
N
-th
root
of
the
fraction-pair
(s
,
s
)
[e.g.,
as
in
Proposition
4.2,
(iii)],
there
exists
a
“morphism”
from
a
suitable
N
-th
root
of
the
fraction-pair
(s
,
s
)
s
N
A
N
−→
⏐
⏐
α
A
s
−→
s
B
N
⏐
⏐
β
N
A
N
−→
B
N
⏐
⏐
⏐
⏐
α
β
B
A
s
−→
B
to
the
given
N
-th
root
of
the
fraction-pair
(s
,
s
),
i.e.,
a
pair
of
commutative
diagrams
s
N
A
N
−→
⏐
⏐
α
N,N
A
N
s
N
−→
s
B
N
⏐
⏐
β
N,N
N
A
N
−→
B
N
⏐
⏐
⏐
α
N,N
⏐
β
N,N
B
N
A
N
s
N
−→
B
N
—
where
α
N,N
(respectively,
β
N,N
)
is
an
isometry
of
Frobenius
degree
N
/N
that
is
compatible
with
α
,
α
(respectively,
β
,
β);
α
N,N
is
of
base-Frobenius
type.
For
instance,
such
a
“morphism”
may
be
constructed
by
extracting
an
“N
/N
-th
root”
[cf.
Proposition
4.2,
(iii)]
of
the
[fraction-pair
determined
by
the]
given
“N
-th
root”
of
the
fraction-pair
(s
,
s
).
Moreover,
this
collection
of
data
induces
a
natural
morphism
H
1
(H
B
N
,
μ
N
(B
N
))
→
H
1
(H
B
N
,
μ
N
(B
N
))
that
maps
κ
f
|
BN
to
the
image
of
κ
f
|
B
in
H
1
(H
B
N
,
μ
N
(B
N
))
[i.e.,
via
the
natural
N
surjection
μ
N
(B
N
)
μ
N
(B
N
)].
In
particular,
by
allowing
N
to
vary,
we
obtain
a
compatible
system
of
roots
of
the
fraction-pair
(s
,
s
)
hence
a
compatible
system
of
Kummer
classes,
which,
by
Proposition
3.2,
(iii),
is
).
sufficient
to
distinguish
f
from
other
elements
of
O
×
(A
birat
We
conclude
our
discussion
of
“general
bi-Kummer
theory”
by
observing
that,
up
to
the
various
indeterminacies
discussed
so
far,
our
constructions
are
entirely
“category-theoretic”:
Theorem
4.4.
(Category-theoreticity
of
Bi-Kummer
Data)
For
i
=
1,
2,
let
p
i
be
a
prime
number;
K
i
a
finite
extension
of
Q
p
i
;
X
i
log
a
smooth
log
orbicurve
over
K
i
;
C
i
a
tempered
Frobenioid,
whose
monoid
type
is
Z,
whose
divisor
monoid
Φ
i
is
perfect
and
non-dilating,
and
whose
base
category
D
i
is
of
the
form
def
D
i
=
B
temp
(X
i
log
)[D
i
]
THE
ÉTALE
THETA
FUNCTION
85
where
D
i
∈
Ob(B
temp
(X
i
log
));
A
,i
∈
Ob(C
i
)
a
Frobenius-trivial
object
such
def
that
A
bs
,i
=
Base(A
,i
)
∈
Ob(D
i
)
is
Galois,
hence
determines
a
normal
open
subgroup
H
,i
⊆
Π
tp
X
i
log
of
the
tempered
fundamental
group
Π
tp
X
i
of
X
i
;
N
∈
N
≥1
.
Suppose
that
∼
Ψ
:
C
1
→
C
2
is
an
equivalence
of
categories
which
induces
[cf.
Theorem
3.7,
(i),
(ii);
Remark
∼
3.7.2;
[Mzk17],
Theorem
3.4,
(v)]
an
equivalence
Ψ
bs
:
D
1
→
D
2
that
maps
A
bs
,1
to
an
isomorph
[cf.
§0]
of
A
bs
.
Then:
,2
(i)
Ψ
maps
isomorphs
of
A
,1
to
isomorphs
of
A
,2
and
H
,1
-ample
objects
∼
to
H
,2
-ample
objects;
Ψ
bs
induces
an
isomorphism
H
,1
→
H
,2
,
which
is
well-
defined
up
to
composition
with
inner
automorphisms
of
Π
tp
X
i
.
∼
(ii)
Ψ
induces
a
1-compatible
equivalence
of
categories
Ψ
birat
:
C
1
birat
→
C
2
birat
.
Moreover,
Ψ
preserves
fraction-pairs,
right
fraction-pairs,
and
left
fraction-
pairs.
Finally,
Ψ
maps
N
-th
roots
of
fraction-pairs
with
domain
isomorphic
to
A
,1
to
N
-th
roots
of
fraction-pairs
with
domain
isomorphic
to
A
,2
.
(iii)
For
i
=
1,
2,
let
A
i
∈
Ob(C
i
)
be
H
,i
-ample;
f
i
∈
O
×
(A
birat
)
an
element
i
fixed
by
the
natural
action
of
H
A
i
[where
“H
A
i
”
is
defined
as
in
Definition
4.1,
(ii),
by
taking
“H
”
to
be
H
,i
];
N
∈
N
≥1
.
Suppose
that
A
1
→
A
2
via
Ψ,
and
that
f
1
→
f
2
via
Ψ
birat
.
Then
A
1
is
(N,
H
,1
,
f
1
)-saturated
if
and
only
if
A
2
is
(N,
H
,2
,
f
2
)-saturated.
Suppose
that,
for
i
=
1,
2,
A
i
is
(N,
H
,i
,
f
i
)-saturated.
Then
the
isomorphism
∼
H
1
(H
A
1
,
μ
N
(A
1
))
→
H
1
(H
A
2
,
μ
N
(A
2
))
maps
κ
f
1
→
κ
f
2
,
i.e.,
is
compatible
with
the
Kummer
classes
of
[Mzk18],
Defini-
tion
2.1,
(ii).
(iv)
For
i
=
1,
2,
let
B
i
∈
Ob(C
i
)
be
an
N
-codomain
of
an
N
-th
root
of
a
fraction-pair
with
domain
isomorphic
to
A
,i
;
write
s
i
-gp
,
s
i
-gp
:
H
B
i
→
Aut
C
i
(B
i
)
for
the
corresponding
bi-Kummer
N
-th
root
[cf.
Proposition
4.3,
(i)]
and
f
i
∈
O
×
(B
i
birat
)
for
the
restriction
to
B
i
[i.e.,
“f
|
B
N
”
in
the
notation
of
Proposition
4.2,
(iii)]
of
the
rational
function
determined
by
the
original
fraction-pair.
Suppose
that
B
1
→
B
2
via
Ψ,
and
that
f
1
→
f
2
via
Ψ
birat
.
Then,
up
to
the
[simultaneous
and
non-simultaneous]
conjugation
operations
discussed
in
Proposition
4.3,
(ii),
the
isomorphisms
∼
H
B
1
→
H
B
2
;
∼
Aut
C
1
(B
1
)
→
Aut
C
2
(B
2
)
86
SHINICHI
MOCHIZUKI
map
s
1
-gp
→
s
2
-gp
,
s
1
-gp
→
s
2
-gp
,
i.e.,
Ψ
is
compatible
with
bi-Kummer
N
-th
roots.
Proof.
First,
we
observe
that,
for
i
=
1,
2,
C
i
is
of
standard
and
isotropic
type,
but
not
of
group-like
type
[cf.
Theorem
3.7,
(i),
(ii)];
the
base
category
D
i
of
C
i
is
slim
[cf.
Remark
3.7.2].
Thus,
Ψ
preserves
pre-steps,
morphisms
of
Frobenius
type,
Frobenius
degrees,
isometries,
and
base-Frobenius
pairs
[cf.
[Mzk17],
Theorem
3.4,
(ii),
(iii);
[Mzk17],
Corollary
5.7,
(i),
(iv)].
In
particular,
[cf.
also
the
existence
of
Ψ
bs
]
Ψ
preserves
Frobenius-trivial
objects.
Since
the
Frobenioid
determined
by
the
Frobenius-trivial
objects
of
C
i
is
of
base-trivial
type
[cf.
[Mzk17],
Theorem
5.1,
(iii)],
these
observations
[together
with
the
existence
of
Ψ
bs
]
imply
the
portion
of
assertion
(i)
concerning
Ψ.
The
portion
of
assertion
(i)
concerning
Ψ
bs
follows
immediately
from
the
theory
of
[quasi-]temperoids
[cf.
[Mzk14],
Proposition
3.2;
[Mzk14],
Theorem
A.4].
Next,
we
consider
assertions
(ii),
(iii).
The
existence
of
Ψ
birat
follows
from
[Mzk17],
Corollary
4.10.
Next,
let
us
recall
that
Ψ
is
compatible
with
the
operation
of
passing
to
the
perfection
[cf.
Theorem
3.7,
(i),
(ii);
[Mzk17],
Theorem
3.4,
(iii)].
In
particular,
it
follows
immediately
from
the
category-theoreticity
of
sets
of
primes
given
in
[Mzk17],
Theorem
4.2,
(ii),
that
Ψ
preserves
base-equivalent
[cf.
the
existence
of
Ψ
bs
!]
pairs
of
pre-steps
whose
Div(−)’s
have
disjoint
supports.
The
remainder
of
assertions
(ii),
(iii)
then
follows
immediately
from
the
observations
thus
far,
the
existence
of
Ψ
bs
,
and
the
“manifestly
category-theoretic
nature”
of
bs-fld
)-saturation”
[cf.
[Mzk18],
Definition
2.2,
(ii);
[Mzk2],
Lemma
1.3.8]
“(N,
H
,i
and
Kummer
classes
[cf.
[Mzk18],
Definition
2.1,
(ii)].
This
completes
the
proof
of
assertions
(ii),
(iii).
Finally,
we
consider
assertion
(iv).
Observe
that
Ψ
preserves
base-Frobenius
pairs
of
Frobenius-trivial
objects
and
[by
assertion
(ii)]
maps
N
-th
roots
of
fraction-
pairs
with
domain
isomorphic
to
A
,1
to
N
-th
roots
of
fraction-pairs
with
domain
isomorphic
to
A
,2
;
Ψ
birat
maps
f
1
→
f
2
.
Thus,
assertion
(iv)
follows
formally
from
Proposition
4.3,
(i),
(ii).
Remark
4.4.1.
Observe
that
one
crucial
difference
between
the
bi-Kummer
theory
for
tempered
Frobenioids
considered
here
and
the
theory
of
[Mzk18],
§2,
in
the
case
of
p-adic
Frobenioids
is
that
in
the
present
case,
there
is
no
reciprocity
map.
This
fact
may
be
regarded
as
a
reflection
of
the
fact
that
the
tempered
group
Π
tp
X
is
not
a
profinite
group
of
cohomological
dimension
2
whose
cohomology
admits
a
duality
theory
of
the
sort
that
G
K
does.
Put
another
way,
although
at
first
glance,
the
Kummer
classes
of
Proposition
4.3,
(iii);
Theorem
4.4,
(iii),
may
appear
to
constitute
a
“purely
[tempered]
fundamental
group-theoretic
presentation”
of
the
Frobenioid-theoretic
rational
functions
under
consideration,
in
fact:
These
Kummer
classes
still
depend
on
a
crucial
piece
of
Frobenioid-theoretic
data
—
namely,
the
cyclotome
“μ
N
(−)”.
Of
course,
if
one
is
only
interested
in
this
cyclotome
up
to
isomorphism
[i.e.,
up
to
multiplication
by
an
element
of
(Z/N
Z)
×
],
then
the
cyclotome
may
be
thought
THE
ÉTALE
THETA
FUNCTION
87
of
as
being
reconstuctible
from
the
cyclotomic
character
of
Π
tp
X
[cf.,
e.g.,
[Mzk2],
Proposition
1.2.1,
(vi)].
On
the
other
hand,
working
with
the
coefficient
cyclo-
tome
up
to
isomorphism
amounts,
in
effect,
[at
the
level
of
rational
functions]
to
×
-powers
—
which
is
typically
unacceptable
working
with
rational
functions
up
to
Z
in
applications
[e.g.,
where
one
wants,
for
instance,
to
specify
the
theta
function,
×
-powers!].
This
technical
issue
of
“rigidity
of
the
not
the
theta
function
up
to
Z
Frobenioid-theoretic
cyclotome”
is,
in
fact,
a
central
theme
of
bi-Kummer
theory
and
will
be
discussed
further
in
§5
below.
88
SHINICHI
MOCHIZUKI
Section
5:
The
Étale
Theta
Function
via
Tempered
Frobenioids
In
the
present
§5,
we
carry
out
the
goal
of
translating
the
scheme-theoretic
constructions
of
§1
into
the
language
of
Frobenioids,
by
applying
the
general
bi-
Kummer
theory
of
§4.
This
translation
yields,
in
particular,
a
Frobenioid-theoretic
construction
of
the
mono-theta
environments
of
§2
[cf.
Theorem
5.10,
(iii)].
In
the
following
discussion,
we
return
to
the
situation
of
Example
3.9.
Suppose
further
that
the
morphism
α
:
A
→
B
of
Example
3.9,
(iv),
is
the
identity
morphism,
and
that
“A”
is
one
of
the
smooth
log
orbicurves
X
log
;
C
log
;
X
log
;
C
log
;
Ẋ
log
;
Ċ
log
;
Ẋ
log
;
Ċ
log
[each
of
which
is
geometrically
connected
over
the
field
K
=
K̈]
of
Definition
2.5,
(i),
(ii),
for
some
odd
integer
l
≥
1.
We
shall
refer
to
the
case
where
“A”
is
X
log
,
C
log
,
Ẋ
log
,
or
Ċ
log
as
the
single
underline
case
and
to
the
case
where
“A”
is
X
log
,
C
log
,
Ẋ
log
,
or
Ċ
log
as
the
double
underline
case.
Now
the
divisor
monoid
Φ
=
Φ
ell
α
def
def
on
D
=
D
α
[cf.
Example
3.9]
determines
a
tempered
Frobenioid
C
of
monoid
type
Z
over
the
base
category
D.
We
would
like
to
apply
the
theory
of
§4
in
the
present
situation.
Thus,
in
the
single
underline
(respectively,
double
underline)
case,
we
take
the
object
A
of
the
theory
of
§4
to
be
the
[Frobenius-trivial]
object
defined
by
the
trivial
line
log
bundle
over
the
object
Ÿ
log
(respectively,
Ÿ
)
of
the
discussion
preceding
Defini-
tion
2.7.
Observe
that
this
A
bs
is
“characteristic”
—
that
is
to
say,
it
is
preserved
by
arbitrary
self-equivalences
of
D
[cf.
Propositions
2.4,
2.6]
—
hence,
in
particular,
Galois.
Let
∼
Ψ:
C
→C
be
a
self-equivalence
of
C;
N
≥
1
an
integer.
Then
it
follows
from
Example
3.9,
(iv),
that
we
have
the
following:
Proposition
5.1.
(Applicability
of
the
General
Theory)
The
Frobenioid
C
is
a
tempered
Frobenioid
of
rationally
standard
type
over
a
slim
base
category
D,
whose
monoid
type
is
Z,
and
whose
divisor
monoid
Φ(−)
is
perfect,
perf-factorial,
non-dilating,
and
cuspidally
pure.
In
particular,
C
and
the
∼
self-equivalence
Ψ
:
C
→
C
satisfy
all
of
the
hypotheses
of
Corollary
3.8,
(i),
(ii),
∼
(iii);
Theorem
4.4
[for
“C
1
”,
“C
2
”,
“Ψ
:
C
1
→
C
2
”].
Remark
5.1.1.
In
Proposition
5.1,
as
well
as
in
the
discussion
of
the
remainder
of
the
present
§5,
we
restrict
our
attention
to
self-equivalences,
instead
of
considering
THE
ÉTALE
THETA
FUNCTION
89
arbitrary
equivalences
between
possibly
distinct
categories,
as
in
Theorem
4.4,
or
“functorial
group-theoretic
algorithms”,
as
in
Corollaries
2.18,
2.19.
We
do
this,
however,
mainly
to
simplify
the
discussion
[in
particular,
the
notation].
That
is
to
say,
the
extension
to
the
case
of
arbitrary
equivalences
between
possibly
distinct
categories
satisfying
similar
hypotheses
is,
for
the
most
part,
immediate.
Moreover,
by
sorting
through
the
various
arguments
used
in
the
proofs
of
the
present
paper,
as
well
as
in
the
proofs
of
the
results
of
[Mzk17],
[Mzk18],
that
are
quoted,
one
concludes
immediately
the
existence
of
“functorial
category-theoretic
algorithms”,
in
the
style
of
Corollaries
2.18,
2.19.
We
leave
the
routine
clerical
details
to
the
interested
reader.
Next,
we
reconsider
the
discussion
of
§1
[where
we
take
the
“N
”
of
§1
to
be
l
·
N
]
from
the
point
of
view
of
the
theory
of
§4.
To
this
end,
let
us
first
observe
log
log
[together
with
the
fact
that
l
·
N
is
divisible
that
by
the
definition
of
Ÿ
,
Z̈
l·N
log
by
l],
it
follows
immediately
that
the
covering
Z̈
l·N
→
X
log
factors
through
the
log
→
X
log
.
Thus,
if
we
pull-back
the
line
bundle
L̈
l·N
on
Ÿ
l·N
[cf.
the
covering
Ÿ
discussion
preceding
Lemma
1.2]
to
Z̈
l·N
[or,
equivalently,
the
line
bundle
L
l·N
on
Y
l·N
,
first
to
Z
l·N
,
and
then
to
Z̈
l·N
]
so
as
to
obtain
a
line
bundle
L̈
l·N
|
Z̈
l·N
on
Z̈
l·N
,
then
the
pull-backs
to
Z̈
l·N
of
(1)
the
section
s
l·N
∈
Γ(Z
l·N
,
L
l·N
|
Z
l·N
)
of
Proposition
1.1,
(i);
(2)
the
theta
trivialization
τ
l·N
∈
Γ(
Ÿ
l·N
,
L̈
l·N
)
of
Lemma
1.2
may
be
interpreted
as
morphisms
V(O
Z̈
l·N
)
→
V(
L̈
l·N
|
Z̈
l·N
)
—
i.e.,
as
morphisms
between
objects
of
C.
Now
we
have
the
following:
Proposition
5.2.
sion:
(Bi-Kummer
Theory)
In
the
notation
of
the
above
discus-
(i)
The
pair
of
morphisms
of
C
determined
by
“s
l·N
”,
“τ
l·N
”
constitutes
an
l
·
N
-th
root
of
a
right
fraction-pair
[cf.
Proposition
4.2,
(iii)]
of
[the
Frobenioid-
theoretic
version
of
the
log-meromorphic
function
constituted
by]
the
theta
func-
tion
Θ̈
of
Proposition
1.4,
or,
alternatively,
an
N
-th
root
of
a
right
fraction-
pair
[cf.
Proposition
4.2,
(iii)]
of
[the
Frobenioid-theoretic
version
of
the
log-
meromorphic
function
constituted
by]
an
l-th
root
of
the
theta
function
Θ̈
[cf.
Remark
4.3.2].
(ii)
The
group
actions
of
Proposition
1.1,
(ii);
Lemma
1.2,
arising
from
“s
l·N
”,
“τ
l·N
”,
respectively,
are
precisely
the
actions
determined
by
the
bi-Kummer
l
·
N
-th
root
[cf.
Proposition
4.3,
(i)]
arising
from
the
l
·
N
-th
root
of
(i).
(iii)
The
Kummer
class
determined
by
the
bi-Kummer
l
·
N
-th
root
of
(ii)
[cf.
Proposition
4.3,
(iii)]
corresponds
precisely
to
the
reduction
modulo
l
·
N
of
the
90
SHINICHI
MOCHIZUKI
class
“η̈
Θ
”
of
Proposition
1.3
—
i.e.,
to
the
“étale
theta
function”
—
relative
to
the
natural
isomorphism
[cf.
Remark
5.2.1
below]
between
“μ
l·N
(−)”
[cf.
Proposition
4.3,
(iii)]
and
Δ
Θ
⊗
(Z/l
·
N
Z)
[cf.
Proposition
1.3].
Similarly,
the
Kummer
class
determined
by
the
bi-Kummer
N
-th
root
of
the
N
-th
root
of
(i)
[cf.
Proposition
4.3,
(iii)]
corresponds
precisely
to
the
reduction
modulo
N
of
the
class
“η̈
Θ
”
of
the
discussion
preceding
Definition
2.7
—
i.e.,
to
an
l-th
root
of
the
“étale
theta
function”
—
relative
to
the
natural
isomorphism
between
“μ
N
(−)
=
l
·
μ
l·N
(−)”
and
(l
·
Δ
Θ
)
⊗
(Z/N
Z).
Proof.
These
assertions
follow
immediately
from
the
definitions.
In
the
case
of
assertion
(i),
we
observe
that
the
“(l
·
N
,
H
,
f
|
A
l·N
)-saturated-ness”
condition
of
Proposition
4.2,
(iii),
follows
immediately
from
the
definition
of
the
field
J
¨
l·N
in
§1.
Remark
5.2.1.
The
natural
isomorphisms
of
Proposition
5.2,
(iii),
constitute
a
scheme-theoretic
ingredient
in
the
otherwise
Frobenioid-theoretic
formulation
of
Proposition
5.2
[cf.
Remark
4.4.1].
The
translation
of
this
final
scheme-theoretic
ingredient
into
category
theory
is
the
topic
of
Proposition
5.5;
Theorems
5.6,
5.7
below.
Next,
we
consider
divisors.
Recall
that
the
special
fiber
of
“
Ÿ”
may
be
de-
scribed
as
an
infinite
chain
of
copies
of
the
projective
line,
joined
to
one
another
at
the
points
“0”
an
“∞”
[cf.,
e.g.,
the
discussion
preceding
Proposition
1.1].
In
particular,
there
is
a
natural
bijection
[cf.
the
discussion
at
the
beginning
of
§1]
∼
Prime(Φ(A
))
ncsp
→
Z
which
is
well-defined,
up
to
the
operations
of
translation
by
an
element
of
Z
and
multiplication
by
±1
on
Z.
Moreover,
there
is
a
natural
surjection
Prime(Φ(A
))
csp
Prime(Φ(A
))
ncsp
[i.e.,
given
by
considering
the
“irreducible
component
of
the
special
fiber
that
con-
tains
the
cusp(s)
determined
by
the
cuspidal
prime”].
Also,
let
us
observe
that
since
all
the
cusps
of
Ÿ
log
arise
from
K̈-rational
points,
it
follows
immediately
that
one
has
the
following
cuspidal
version
of
Remark
3.8.2:
If
p,
q
are
cuspidal
primes
of
Φ(A
),
then
there
is
a
natural
isomorphism
between
“primary
components”
∼
Φ(A
)
p
→
Φ(A
)
q
—
determined
by
identifying
the
elements
on
each
side
that
arise
from
[scheme-
theoretic]
prime
log-divisors.
∼
Next,
let
us
choose
an
isomorphism
Ψ(A
)
→
A
of
C
[cf.
Proposition
5.1;
Theorem
4.4,
(i)].
Let
us
write
∼
Ψ
Φ
A
:
Φ(A
)
→
Φ(A
)
THE
ÉTALE
THETA
FUNCTION
91
for
the
automorphism
of
the
monoid
Φ(A
)
obtained
by
composing
the
isomor-
∼
phism
of
divisor
monoids
induced
by
Ψ
:
C
→
C
[cf.
Corollary
3.8,
(iii);
[Mzk17],
∼
Theorem
4.9]
with
the
isomorphism
Φ(Ψ(A
))
→
Φ(A
)
induced
by
the
chosen
∼
isomorphism
Ψ(A
)
→
A
.
Proposition
5.3.
(Category-theoreticity
of
the
Geometry
of
Divisors)
In
the
notation
of
the
above
discussion,
Ψ
Φ
A
preserves
the
following
objects:
(i)
non-cuspidal
and
cuspidal
elements;
(ii)
the
natural
isomorphisms
between
distinct
non-cuspidal
primary
components
of
Φ(A
)
[cf.
Remark
3.8.2];
(iii)
the
natural
isomorphisms
[described
in
the
discussion
above]
between
distinct
cuspidal
primary
components
of
Φ(A
);
(iv)
the
natural
surjection
Prime(Φ(A
))
csp
Prime(Φ(A
))
ncsp
;
∼
(v)
the
natural
bijection
Prime(Φ(A
))
ncsp
→
Z
[up
to
translation
by
an
element
of
Z
and
multiplication
by
±1];
(vi)
the
Aut
C
(A
)-orbit
of
the
divisor
of
zeroes
and
poles
∈
Φ(A
)
gp
of
[the
Frobenioid-theoretic
version
of
the
log-meromorphic
function
constituted
by]
the
theta
function
Θ̈
of
Proposition
1.4.
Proof.
The
preservation
of
(i)
(respectively,
(ii))
follows
immediately
from
Corol-
lary
3.8,
(iii)
(respectively,
Remark
3.8.2).
To
verify
the
preservation
of
the
remain-
ing
objects,
it
suffices
to
consider
the
well-known
“intersection
theory
of
divisors
supported
on
the
chain
of
copies
of
the
projective
line”
that
constitutes
the
special
divisor
of
“
Ÿ”
as
follows:
Let
us
refer
to
pairs
of
elements
of
Φ(A
)
gp
whose
difference
lies
in
the
image
of
the
birational
function
monoid
of
the
Frobenioid
C
as
linearly
equivalent.
Let
us
refer
to
an
element
∈
Φ(A
)
gp
which
is
linearly
equivalent
to
0
as
principal.
Let
us
refer
to
as
cuspidally
minimal
any
cuspidal
element
of
Φ(A
)
gp
whose
support
[cf.
[Mzk17],
Definition
2.4,
(i),
(d)]
is
a
finite
set
of
minimal
cardinality
among
the
cardinalities
of
supports
of
cuspidal
elements
of
Φ(A
)
gp
which
are
linearly
equivalent
to
the
given
element.
Now
the
preservation
of
(iv)
follows
by
considering
the
support
of
primary
cuspidal
elements
a
∈
Φ(A
)
such
that,
for
some
b
∈
Φ(A
)
csp
which
is
co-prime
to
a
[i.e.,
a,
b
have
disjoint
supports],
b
−
a
is
cuspidally
minimal
and
linearly
equivalent
to
a
primary
non-cuspidal
element
n
∈
Φ(A
).
That
is
to
say,
[by
the
well-known
intersection
theory
of
divisors
supported
on
the
chain
of
copies
of
the
projective
line]
in
this
situation,
n
is
linearly
equivalent
to
some
element
∈
Φ(A
)
gp
of
the
form
n
1
+
n
2
−
a,
where
n
1
,
n
2
∈
Φ(A
)
csp
are
primary
cuspidal
elements
that
map,
respectively,
via
the
natural
surjection
of
(iv)
to
the
two
non-cuspidal
primes
that
are
adjacent
[in
the
“chain
of
copies
of
the
projective
line”]
to
the
non-
cuspidal
prime
determined
by
n.
Moreover,
relative
to
the
isomorphisms
of
(iii),
the
92
SHINICHI
MOCHIZUKI
multiplicities
of
n
1
,
n
2
are
equal
to
each
other
as
well
as
to
half
the
multiplicity
of
a.
Thus,
the
natural
isomorphisms
of
(iii)
may
be
obtained
by
applying
the
natural
isomorphisms
of
(ii)
to
the
“n”
that
occur
for
various
“a”;
the
natural
surjection
of
(iv)
is
obtained
by
mapping
the
prime
determined
by
a
to
the
prime
determined
by
n.
To
verify
the
preservation
of
(v),
it
suffices
to
show
that
the
relation
of
ad-
jacency
[in
the
“chain
of
copies
of
the
projective
line”]
between
elements
p,
q
∈
Φ(A
)
ncsp
is
preserved.
But
this
follows
[again
from
the
well-known
intersection
theory
of
divisors
supported
on
the
chain
of
copies
of
the
projective
line]
by
ob-
serving
that
if
a
∈
p,
b
∈
q
correspond
via
the
natural
isomorphisms
of
(ii),
then
p,
q
are
adjacent
(respectively,
not
adjacent)
if
and
only
if
every
cuspidally
minimal
c
∈
Φ(A
)
gp
which
is
linearly
equivalent
to
a
+
b
has
support
of
cardinality
4
(re-
spectively,
5
or
6).
[Here,
the
numbers
“4”,
“5”,
“6”
correspond
to
the
number
of
non-cuspidal
primes
that
are
either
contained
in
or
adjacent
to
a
prime
contained
in
the
support
of
a
+
b.]
Finally,
the
preservation
of
(vi),
at
least
up
to
Q
>0
-multiples,
follows
immedi-
ately
by
considering
the
principal
divisors
∈
Φ(A
)
gp
in
light
of
the
preservation
of
(i),
(ii),
(iii),
and
(v)
[cf.
the
description
of
the
divisor
of
zeroes
and
poles
of
Θ̈
in
Proposition
1.4,
(i)];
to
eliminate
the
indeterminacy
with
respect
to
Q
>0
-
multiples,
it
suffices
to
consider
the
zero
divisor
[cf.
Remark
3.8.2]
of
a
generator
of
O
(A
)/O
×
(A
)
∼
=
Z
≥0
[cf.
Proposition
3.4,
(ii)].
Next,
let
us
recall
the
characteristic
[cf.
Propositions
2.4,
2.6]
subquotients
tp
Θ
Π
tp
X
(Π
X
)
;
Θ
l
·
Δ
Θ
⊆
(Π
tp
X
)
of
the
discussion
at
the
beginning
of
§1.
Thus,
for
D
∈
Ob(D),
these
subquotients
determine
subquotients
Aut
D
(D)
Aut
Θ
D
(D);
(l
·
Δ
Θ
)
D
⊆
Aut
Θ
D
(D)
which
are
preserved
by
arbitrary
self-equivalences
of
D
[cf.
Propositions
2.4,
2.6].
def
If
S
∈
Ob(C),
then
we
shall
write
(l
·
Δ
Θ
)
S
=
(l
·
Δ
Θ
)
S
bs
.
Definition
5.4.
We
shall
say
that
S
∈
Ob(C)
is
(l,
N
)-theta-saturated
if
the
following
conditions
are
satisfied:
(a)
S
is
μ
l·N
-saturated
[cf.
[Mzk18],
Definition
2.1,
(i)];
(b)
(l
·
Δ
Θ
)
S
⊗
Z/N
Z
is
of
cardinality
N
.
Now
we
may
begin
to
address
the
issue
discussed
in
Remark
5.2.1:
Proposition
5.5.
(Frobenioid-theoretic
Cyclotomic
Rigidity)
In
the
no-
tation
of
the
above
discussion,
the
second
Kummer
class
of
Proposition
5.2,
(iii),
determines
an
isomorphism
∼
(l
·
Δ
Θ
)
S
⊗
Z/N
Z
→
μ
N
(S)
(=
l
·
μ
l·N
(S))
THE
ÉTALE
THETA
FUNCTION
93
for
all
(l,
N
)-theta-saturated
S
∈
Ob(C).
This
isomorphism
is
functorial
with
respect
to
the
subcategory
of
C
determined
by
the
linear
morphisms
of
(l,
N
)-
theta-saturated
objects.
Proof.
Indeed,
if
S
∈
Ob(C)
is
an
l
·
N
-codomain
of
an
l
·
N
-th
root
of
a
right
fraction-pair
of
Θ̈
[i.e.,
as
discussed
in
Proposition
5.2,
(i)],
then
it
follows
from
the
detailed
description
of
the
“étale
theta
class”
in
Proposition
1.3
that
the
resulting
Kummer
class
[cf.
Proposition
5.2,
(iii)]
determines
an
isomorphism
∼
(l
·
Δ
Θ
)
S
⊗
Z/N
Z
→
μ
N
(S
)
which
is
manifestly
“functorial”
[in
the
evident
sense,
with
respect
to
l
·
N
-codomains
of
l
·
N
-th
roots
of
right
fraction-pairs
of
Θ̈].
Thus,
we
may
transport
this
isomor-
phism
from
S
to
an
arbitrary
(l,
N
)-theta-saturated
S
∈
Ob(C)
by
means
of
linear
morphisms
S
→
S,
S
→
S
[of
C
—
cf.
[Mzk17],
Definition
1.3,
(i),
(b)],
which
induce
isomorphisms
∼
(l
·
Δ
Θ
)
S
⊗
Z/N
Z
→
(l
·
Δ
Θ
)
S
⊗
Z/N
Z;
∼
(l
·
Δ
Θ
)
S
⊗
Z/N
Z
→
(l
·
Δ
Θ
)
S
⊗
Z/N
Z;
∼
μ
N
(S)
→
μ
N
(S
);
∼
μ
N
(S
)
→
μ
N
(S
)
∼
—
hence
also
a
[functorial]
isomorphism
(l
·
Δ
Θ
)
S
⊗
Z/N
Z
→
μ
N
(S),
which
is
independent
of
the
choice
of
S
,
S
and
the
linear
morphisms
S
→
S,
S
→
S
[precisely
because
of
the
original
“functoriality”
of
the
isomorphism
for
S
].
Theorem
5.6.
(Category-theoreticity
of
Frobenioid-theoretic
Cyclo-
tomic
Rigidity)
Write
C
for
the
tempered
Frobenioid
of
monoid
type
Z
de-
def
def
termined
by
the
divisor
monoid
Φ
=
Φ
ell
α
on
the
base
category
D
=
D
α
of
Example
3.9,
(iv),
where
we
take
the
morphism
α
:
A
→
B
of
Example
3.9,
(iv),
to
be
the
identity
morphism
and
“A”
to
be
one
of
the
smooth
log
orbicurves
X
log
;
C
log
;
X
log
;
C
log
;
Ẋ
log
;
Ċ
log
;
Ẋ
log
;
Ċ
log
[each
of
which
is
geometrically
connected
over
the
field
K
=
K̈]
of
Definition
2.5,
(i),
(ii),
for
some
odd
integer
l
≥
1.
Let
∼
Ψ:
C
→C
be
a
self-equivalence
of
C;
N
≥
1
an
integer.
Then
Ψ
preserves
the
(l,
N
)-
theta-saturated
objects,
as
well
as
the
natural
isomorphism
∼
(l
·
Δ
Θ
)
S
⊗
Z/N
Z
→
μ
N
(S)
(=
l
·
μ
l·N
(S))
[for
(l,
N
)-theta-saturated
S
∈
Ob(C)]
of
Proposition
5.5
[i.e.,
Ψ
transports
this
isomorphism
for
S
to
the
corresponding
isomorphism
for
Ψ(S)].
94
SHINICHI
MOCHIZUKI
Proof.
Indeed,
by
Proposition
5.1;
[Mzk17],
Theorem
3.4,
(iv),
(v),
it
follows
that
∼
Ψ
preserves
“O
×
(−)”
and
induces
a
1-compatible
equivalence
Ψ
bs
:
D
→
D,
hence,
[cf.
[Mzk14],
Proposition
3.2]
an
outer
automorphism
of
the
tempered
fundamental
group
of
the
smooth
orbicurve
in
question.
In
particular,
it
follows
from
Proposi-
tions
2.4,
2.6
that
Ψ
preserves
“(l
·
Δ
Θ
)
(−)
”.
Thus,
it
follows
immediately
that
Ψ
preserves
the
(l,
N
)-theta-saturated
objects.
Next,
let
S
2
∈
Ob(C)
be
an
(l,
N
)-theta-saturated
l
·
N
-codomain
of
an
l
·
N
-th
root
of
a
right
fraction-pair
of
Θ̈
[i.e.,
as
discussed
in
Proposition
5.2,
(i)]
such
that
S
2
bs
is
“characteristic”
[i.e.,
its
isomorphism
class
is
preserved
by
arbitrary
self-equivalences
of
D];
write
s
,
s
:
S
1
→
S
2
for
the
pair
of
base-equivalent
pre-steps
that
appear
in
this
l
·
N
-th
root
[so
S
1
is
Frobenius-trivial].
Now
since
Ψ
preserves
pre-steps
and
Frobenius-trivial
objects
∼
and
induces
a
1-compatible
self-equivalence
Ψ
bs
:
D
→
D
[cf.
the
proof
of
Theorem
4.4],
it
follows
that
Ψ
maps
this
pair
of
base-equivalent
pre-steps
to
a
pair
of
base-
equivalent
pre-steps
t
,
t
:
T
1
→
T
2
such
that
T
1
is
Frobenius-trivial,
hence
isomorphic
to
S
1
[cf.
our
assumption
that
S
1
bs
∼
=
S
2
bs
is
characteristic;
[Mzk17],
Theorem
5.1,
(iii)]
.
Moreover,
since
Ψ
[essen-
tially]
preserves
the
divisor
of
zeroes
and
poles
of
Θ̈
[cf.
Proposition
5.3,
(vi)],
it
follows
[cf.
the
equivalences
of
categories
involving
pre-steps
of
[Mzk17],
Definition
∼
∼
1.3,
(iii),
(d)]
that
there
exist
isomorphisms
γ
1
:
S
1
→
T
1
,
γ
2
:
S
2
→
T
2
,
u
∈
O
×
(T
2
)
such
that
γ
2
◦
s
=
t
◦
γ
1
,
u
◦
γ
2
◦
s
=
t
◦
γ
1
.
Thus,
by
forming
the
resulting
group
homomorphisms
s
-gp
,
s
-gp
:
H
S
2
→
Aut
C
(S
2
);
t
-gp
,
t
-gp
:
H
T
2
→
Aut
C
(T
2
)
[cf.
Proposition
4.3,
(i)]
and
Kummer
classes
[cf.
Proposition
4.3,
(iii)],
and
ob-
serving
that
the
Kummer
class
of
the
“constant
function”
u
does
not
affect
the
restriction
of
the
resulting
Kummer
classes
to
(l
·
Δ
Θ
)
S
2
,
(l
·
Δ
Θ
)
T
2
,
we
conclude
∼
that
Ψ
does
indeed
transport
the
natural
isomorphism
(l
·Δ
Θ
)
S
2
⊗Z/N
Z
→
μ
N
(S
2
)
of
Proposition
5.5
to
the
corresponding
isomorphism
for
T
2
.
Moreover,
by
the
con-
struction
applied
in
the
proof
of
Proposition
5.5,
this
preservation
of
the
natural
isomorphism
of
Proposition
5.5
in
the
specific
case
of
S
2
is
sufficient
to
imply
the
preservation
of
the
natural
isomorphism
of
Proposition
5.5
for
arbitrary
(l,
N
)-
theta-saturated
S
∈
Ob(C).
This
completes
the
proof
of
Theorem
5.6.
Theorem
5.7.
(Category-theoreticity
of
the
Frobenioid-theoretic
Theta
Function)
In
the
notation
of
Theorem
5.6,
Ψ
preserves
right
fraction-pairs
of
[the
Frobenioid-theoretic
version
of
the
log-meromorphic
function
constituted
by]
an
l-th
root
of
the
theta
function
√
Θ̈(
−1)
−1
·
Θ̈
THE
ÉTALE
THETA
FUNCTION
95
of
Proposition
1.4
[normalized
so
as
to
be
“of
standard
type”],
up
to
possible
multiplication
by
a
2l-th
root
of
unity
and
possible
translation
by
an
element
log
of
Z
(
∼
=
Gal(
Ÿ
/Ẋ))
[when
“A”
arises
from
X
log
,
Ẋ
]
or
l
·
Z
[when
“A”
arises
from
C
log
,
Ċ
log
,
X
log
,
Ẋ
log
,
C
log
,
Ċ
log
].
Proof.
Indeed,
this
follows
immediately
by
considering
compatible
systems
as
in
Remark
4.3.2
and
applying
the
theory
of
the
rigidity
of
the
étale
theta
function
[cf.
Corollary
2.8,
(i)],
to
the
Kummer
classes
of
Proposition
5.2,
(iii)
[cf.
also
Theorem
4.4,
(iii);
Proposition
5.3,
(vi)],
above,
in
light
of
the
crucial
isomorphisms
∼
(l·Δ
Θ
)
S
⊗Z/N
Z
→
μ
N
(S)
of
Proposition
5.5,
which,
by
Theorem
5.6,
are
preserved
by
Ψ.
Remark
5.7.1.
Note
that
the
“rigidity
up
to
possible
multiplication
by
a
2l-th
root
of
unity”
asserted
in
Theorem
5.7
is
substantially
stronger
than
[what
was
in
effect]
the
preservation
of
Θ̈
up
to
multiplication
by
an
arbitrary
constant
function
∈
O
×
(−)
[cf.
the
“u”
appearing
in
the
proof
of
Theorem
5.6],
that
was
observed
by
considering
divisors
[i.e.,
Proposition
5.3,
(vi)]
in
the
proof
of
Theorem
5.6.
Next,
we
consider
theta
environments.
For
the
remainder
of
the
present
§5,
we
suppose
that:
“A”
arises
from
X
log
.
Write
s
N
,
s
N
:
A
N
→
B
N
for
the
pair
of
base-equivalent
morphisms
of
C
determined
by
the
sections
“s
l·N
”,
“τ
l·N
”
of
the
discussion
preceding
Proposition
5.2
[so
s
N
,
s
N
constitute
an
N
-th
root
of
a
right
fraction-pair
of
an
l-th
root
of
the
theta
function
Θ̈
—
cf.
Proposition
5.2,
(i)].
Note
that
the
zero
divisor
Div(s
N
)
∈
Φ(A
N
)
of
s
N
descends
[relative
to
the
unique
morphism
A
bs
→
A
in
D]
to
Φ(A)
[cf.
Proposition
1.4,
(i)];
in
particular,
it
bs
follows
that
B
N
is
Aut-ample.
Thus,
the
group
homomorphism
s
trv
N
:
Aut
D
(A
N
)
→
Aut
C
(A
N
)
arising
from
a
base-Frobenius
pair
of
A
N
[cf.
Proposition
5.1;
Theorem
3.7,
(i);
[Mzk17],
Proposition
5.6],
which
is
completely
determined
[cf.
Proposition
5.1;
Theorem
3.7,
(i);
[Mzk17],
Proposition
5.6]
up
to
conjugation
by
an
element
of
O
×
(A
N
),
determines
unique
group
homomorphisms
bs
s
N
-gp
:
Aut
D
(B
N
)
→
Aut
C
(B
N
);
s
N
-gp
:
H
B
N
→
Aut
C
(B
N
)
∼
∼
bs
such
that,
relative
to
the
isomorphisms
Aut
D
(A
bs
N
)
→
Aut
D
(B
N
),
H
A
N
→
H
B
N
determined
by
the
[base-equivalent!]
pair
of
morphisms
s
N
,
s
N
,
we
have
s
N
-gp
(g)
◦
s
N
=
s
N
◦
(s
trv
bs
)
)(g);
N
|
Aut
D
(B
N
s
N
-gp
(h)
◦
s
N
=
s
N
◦
(s
trv
N
|
H
BN
)(h)
bs
),
h
∈
H
B
N
[cf.
Propositions
4.3,
(i);
5.2,
(ii)].
Write
for
all
g
∈
Aut
D
(B
N
def
E
N
=
s
N
-gp
(Im(Π
tp
Y
))
·
μ
N
(B
N
)
⊆
Aut
C
(B
N
)
96
SHINICHI
MOCHIZUKI
tp
tp
—
where
Im(Π
tp
Y
)
denotes
the
image
of
Π
Y
⊆
Π
X
via
the
natural
outer
homomor-
bs
phism
Π
tp
X
Aut
D
(B
N
)
[cf.
Definition
4.1,
(ii)].
Lemma
5.8.
discussion,
write
(Conjugation
by
Constants)
In
the
notation
of
the
above
birat
(K
×
)
1/N
⊆
O
×
(B
N
)
for
the
subgroup
of
elements
whose
N
-th
power
lies
in
the
image
of
the
natural
×
1/N
def
×
1/N
birat
);
(O
K
)
=
(K
×
)
1/N
O
×
(B
N
).
Then
(O
K
)
inclusion
K
×
→
O
×
(B
N
×
is
equal
to
the
set
of
elements
of
O
(B
N
)
that
normalize
the
subgroup
E
N
⊆
Aut
C
(B
N
).
In
particular,
we
have
a
natural
outer
action
of
∼
×
1/N
×
)
/μ
N
(B
N
)
→
O
K
(O
K
∼
on
E
N
;
this
outer
action
extends
to
an
outer
action
of
(K
×
)
1/N
/μ
N
(B
N
)
→
K
×
on
E
N
.
Proof.
Indeed,
since
Y
is
geometrically
connected
over
K,
it
follows
immediately
that
the
set
of
elements
of
O
×
(B
N
)
that
normalize
the
subgroup
E
N
⊆
Aut
C
(B
N
)
is
equal
to
the
set
of
elements
on
which
Π
tp
Y
[i.e.,
G
K
,
via
the
natural
surjection
Π
tp
Y
G
K
]
acts
via
multiplication
by
an
element
of
μ
N
(B
N
).
But
this
last
set
is
×
1/N
)
.
easily
seen
to
coincide
with
(O
K
Lemma
5.9.
(First
Properties
of
Frobenioid-theoretic
Theta
Environ-
ments)
In
the
notation
of
the
above
discussion:
(i)
s
N
-gp
|
H
BN
,
s
N
-gp
factor
through
E
N
.
(ii)
We
have
a
natural
exact
sequence
1
→
μ
N
(B
N
)
→
E
N
→
Im(Π
tp
Y
)
→
1
tp
tp
—
where
Im(Π
tp
Y
)
denotes
the
image
of
Π
Y
⊆
Π
X
via
the
natural
outer
homomor-
bs
phism
Π
tp
X
Aut
D
(B
N
)
[cf.
Definition
4.1,
(ii)].
(iii)
We
have
a
natural
outer
action
∼
tp
l
·
Z
→
Π
tp
X
/Π
Y
→
Out(E
N
)
determined
by
conjugating
via
the
composite
of
the
natural
outer
homomorphism
-gp
bs
bs
Π
tp
:
Aut
D
(B
N
)
→
Aut
C
(B
N
).
X
Aut
D
(B
N
)
with
s
N
(iv)
Write
def
tp
E
Π
N
=
E
N
×
Im(Π
tp
)
Π
Y
Y
THE
ÉTALE
THETA
FUNCTION
97
Π
—
where
Im(Π
tp
Y
)
is
as
in
(ii);
E
N
,
E
N
are
equipped
with
the
evident
topologies;
tp
the
homomorphism
Π
tp
Y
Im(Π
Y
)
is
well-defined
up
to
conjugation
by
an
element
tp
of
Π
tp
X
.
Then
the
natural
inclusions
μ
N
(B
N
)
→
E
N
,
Im(Π
Y
)
⊆
E
N
determine
an
isomorphism
of
topological
groups
∼
tp
E
Π
N
→
Π
Y
[μ
N
]
which
is
an
isomorphism
of
mod
N
bi-theta
environments
with
respect
to
the
model
bi-theta
environment
structure
of
Definition
2.13,
(ii),
on
Π
tp
Y
[μ
N
]
and
the
mod
N
bi-theta
environment
structure
on
E
Π
N
determined
by
the
subgroup
of
)
generated
by
the
natural
outer
actions
of
l
·
Z
[cf.
(iii)],
K
×
[cf.
Lemma
Out(E
Π
N
5.8]
on
E
N
,
together
with
the
μ
N
-conjugacy
classes
of
subgroups
given
by
the
images
of
the
homomorphisms
s
N
-Π
,
s
N
-Π
:
Π
tp
→
E
Π
N
Ÿ
arising
from
s
N
-gp
,
s
N
-gp
[cf.
(i)].
In
particular,
omitting
the
homomorphism
s
N
-Π
yields
a
mod
N
mono-theta
environment.
(v)
In
the
situation
of
(iv),
the
cyclotomic
rigidity
isomorphism
arising
from
the
theory
of
§2
[cf.
Corollary
2.19,
(i)]
coincides
with
the
Frobenioid-theoretic
isomorphism
of
Proposition
5.5
[where
we
take
“S”
to
be
B
N
].
Proof.
Immediate
from
the
definitions
[with
regard
to
assertion
(iii),
cf.
also
Proposition
5.2,
(ii)].
We
are
now
ready
to
state
our
main
result
relating
the
theory
of
theta
envi-
ronments
of
§2
to
the
theory
of
tempered
Frobenioids
discussed
in
§3,
§4,
and
the
present
§5.
Theorem
5.10.
(Category-theoreticity
of
Frobenioid-theoretic
Theta
Environments)
In
the
notation
of
Theorem
5.6,
suppose
further
that
“A”
arises
from
X
log
;
write
s
N
,
s
N
:
A
N
→
B
N
for
the
pair
of
base-equivalent
morphisms
of
C
determined
by
the
sections
“s
l·N
”,
“τ
l·N
”
of
the
discussion
preceding
Proposition
5.2
—
so
s
N
,
s
N
constitute
an
N
-
th
root
of
a
right
fraction-pair
of
an
l-th
root
of
the
theta
function
Θ̈
[cf.
Proposition
5.2,
(i)];
E
N
⊆
Aut
C
(B
N
)
for
the
subgroup
defined
in
the
discussion
preceding
Lemma
5.8;
def
tp
E
Π
N
=
E
N
×
Im(Π
tp
)
Π
Y
Y
98
SHINICHI
MOCHIZUKI
for
the
topological
group
defined
in
Lemma
5.9,
(iv);
:
E
Π
N
→
Aut
C
(B
N
)
for
the
μ
N
-outer
homomorphism
[i.e.,
homomorphism
considered
up
to
com-
tp
position
with
an
inner
automorphism
defined
by
an
element
of
Ker(E
Π
N
Π
Y
)
or
μ
N
(B
N
)
⊆
Aut
C
(B
N
)]
determined
by
the
natural
projection
E
Π
N
→
E
N
(⊆
Aut
C
(B
N
)).
Then:
∼
B
N
.
(i)
The
self-equivalence
Ψ
:
C
→
C
preserves
the
isomorphism
classes
of
A
N
,
∼
(ii)
Let
β
be
an
isomorphism
β
:
Ψ(B
N
)
→
B
N
[cf.
(i)];
write
∼
Ψ
Aut
:
Aut
C
(B
N
)
→
Aut
C
(B
N
);
∼
birat
birat
Ψ
birat
)
→
Aut
C
birat
(B
N
)
Aut
:
Aut
C
birat
(B
N
for
the
automorphisms
determined
by
applying
Ψ
followed
by
conjugation
by
β.
Then
Ψ
Aut
,
Ψ
birat
Aut
preserve
O
×
(B
N
);
×
1/N
(O
K
)
⊆
O
×
(B
N
);
birat
O
×
(B
N
);
birat
(K
×
)
1/N
⊆
O
×
(B
N
)
Im(s
N
-gp
);
Im(s
N
-gp
)
and
map
the
data
E
N
(⊆
Aut
C
(B
N
));
—
where
“Im(−)”
denotes
the
image
of
the
homomorphism
in
parentheses
—
to
data
δ
1
·
E
N
·
δ
1
−1
(⊆
Aut
C
(B
N
));
δ
1
·
Im(s
N
-gp
)
·
δ
1
−1
;
δ
1
·
δ
2
·
δ
3
·
Im(s
N
-gp
)
·
δ
3
−1
·
δ
2
−1
·
δ
1
−1
×
1/N
for
some
δ
1
∈
O
×
(B
N
),
some
δ
2
∈
μ
2l·N
(B
N
)
(O
K
)
(⊆
O
×
(B
N
)),
and
some
-gp
bs
δ
3
∈
s
N
(Aut
D
(B
N
)).
(iii)
The
operation
of
applying
Ψ
followed
by
conjugation
by
β
preserves
the
Aut
C
(B
N
)-orbit
of
:
E
Π
N
→
Aut
C
(B
N
),
in
a
fashion
which
is
compati-
-Π
discussed
ble
with
the
mono-theta
environment
structure
on
E
Π
N
involving
s
N
in
Lemma
5.9,
(iv).
More
precisely:
there
exists
a
commutative
diagram
E
Π
N
⏐
⏐
Aut
C
(B
N
)
γ
−→
Ψ
Aut
−→
E
Π
N
⏐
⏐
κ◦
Aut
C
(B
N
)
—
where
κ
is
an
inner
automorphism
of
Aut
C
(B
N
);
γ
is
an
automorphism
of
topological
groups
which
determines
an
automorphism
of
mono-theta
envi-
tp
ronments
and
is
compatible
with
the
Π
tp
X
-conjugacy
class
of
automorphisms
of
Π
Y
induced
by
Ψ
bs
[cf.
Theorem
4.4].
THE
ÉTALE
THETA
FUNCTION
99
Proof.
First,
we
observe
that
by
Proposition
5.1,
the
hypotheses
of
Theorem
4.4
are
satisfied.
Next,
we
consider
assertion
(i).
Since
Ψ
preserves
Frobenius-trivial
objects
[cf.
the
proof
of
Theorem
4.4],
to
show
that
Ψ
preserves
the
isomorphism
∼
class
of
A
N
,
it
suffices
to
show
that
the
equivalence
Ψ
bs
:
D
→
D
[cf.
Theorem
4.4]
induced
by
Ψ
preserves
the
isomorphism
class
of
the
objects
of
D
determined
log
log
”,
“
Ÿ
”;
but
this
follows
immediately
[in
light
of
the
definitions
of
the
by
“
Z̈
l·N
various
tempered
coverings
involved]
from
Proposition
2.4.
Now
the
fact
that
Ψ
preserves
the
isomorphism
class
of
B
N
follows
immediately
from
Proposition
5.3,
(vi).
This
completes
the
proof
of
assertion
(i).
Next,
we
consider
assertion
(ii).
First,
we
observe
that
it
follows
from
the
existence
of
Ψ
bs
and
the
fact
that
Ψ
preserves
pre-steps
[cf.
the
proof
of
Theorem
4.4]
that
Ψ
preserves
“O
(−)”,
as
well
as
base-equivalent
pairs
of
pre-steps.
In
par-
ticular,
it
follows
that
applying
Ψ
followed
by
conjugation
by
β
preserves
O
(B
N
),
birat
).
Thus,
[cf.
Proposition
3.4,
(ii)]
we
conclude
[by
considering
O
×
(B
N
),
O
×
(B
N
Galois-,
i.e.,
Aut
C
(B
N
)-invariants]
that
Ψ
birat
Aut
preserves
the
image
of
the
natural
×
1/N
×
×
birat
)
.
Finally,
the
portion
inclusion
K
→
O
(B
N
),
hence
also
(K
×
)
1/N
,
(O
K
-gp
-gp
of
assertion
(ii)
concerning
Im(s
N
),
Im(s
N
),
E
N
follows
by
observing
that
Ψ
preserves
N
-th
roots
of
fraction
pairs
[cf.
Theorem
4.4,
(ii)],
together
with
the
corresponding
bi-Kummer
N
-th
roots
[cf.
Theorem
4.4,
(iv)],
for
[the
Frobenioid-
theoretic
version
of
the
log-meromorphic
function
constituted
by]
l-th
roots
of
the
theta
function
[i.e.,
up
to
the
indeterminacies
discussed
in
the
statement
of
Theorem
5.7
—
cf.
Theorem
5.7].
This
completes
the
proof
of
assertion
(ii).
Finally,
we
consider
assertion
(iii).
First,
observe
that
it
follows
from
the
existence
of
κ
that,
in
the
following
argument,
we
may
treat
as
a
single
fixed
homomorphism,
rather
than
just
a
“μ
N
-outer
homomorphism”.
Next,
we
observe
that
by
taking
κ
to
be
the
inner
automorphism
of
Aut
C
(B
N
)
determined
by
con-
jugating
by
the
element
δ
1
·
δ
2
·
δ
3
∈
Aut
C
(B
N
)
of
assertion
(ii),
we
may
assume
that
the
restrictions
of
Ψ
Aut
◦
,
κ
◦
to
the
image
of
s
N
-Π
:
Π
tp
→
E
Π
N
coincide,
Ÿ
that
extends
up
to
composition
with
an
automorphism
of
the
topological
group
Π
tp
Ÿ
tp
to
an
automorphism
of
the
topological
group
Π
tp
X
[which
lies
in
the
Π
X
-conjugacy
bs
class
of
automorphisms
of
Π
tp
Y
induced
by
Ψ
].
Thus,
by
applying
Corollary
2.18,
(iv),
it
follows
that
we
may
choose
γ
so
that
the
restrictions
of
Ψ
Aut
◦
,
κ
◦
◦
γ
to
the
image
of
s
N
-Π
coincide
[precisely].
Moreover,
by
applying
the
compatible
[cf.
Lemma
5.9,
(v)],
category/group-theoretic
[cf.
Theorem
5.6;
Corollary
2.19,
(i)]
cyclotomic
rigidity
isomorphisms
of
Proposition
5.5
and
Corollary
2.19,
(i)
[cf.
also
Remark
2.19.3],
it
follows
that
the
restrictions
of
Ψ
Aut
◦
,
κ
◦
◦
γ
to
tp
Ker(E
Π
N
Π
Y
)
coincide.
Thus,
we
conclude
that
the
restrictions
of
Ψ
Aut
◦
,
∼
Π
κ
◦
◦
γ
to
Π
tp
[μ
N
]
⊆
Π
tp
Y
[μ
N
]
→
E
N
[where
we
apply
the
isomorphism
of
Lemma
Ÿ
tp
5.9,
(iv)]
coincide.
Since
[Π
tp
Y
:
Π
Ÿ
]
=
2,
it
thus
follows
that
the
difference
between
Ψ
Aut
◦
,
κ
◦
◦
γ
determines
a
cohomology
class
[cf.
the
proof
of
Corollary
2.18,
(iv)]
tp
×
∼
×
G
K
∼
∈
H
1
(Π
tp
)
=
H
1
(Z/2Z,
O
K
)
=
μ
2
(B
N
)
Y
/Π
Ÿ
,
O
(B
N
)
100
SHINICHI
MOCHIZUKI
—
where
the
superscript
“G
K
”
denotes
the
subgroup
of
G
K
-invariants;
we
apply
Proposition
3.4,
(ii),
and
recall
that
K̈
=
K.
Thus,
by
composing
γ
with
an
appropriate
order
two
automorphism
of
mono-theta
environments
[cf.
Corollary
2.18,
(iv),
and
its
proof],
we
obtain
a
γ
such
that
Ψ
Aut
◦
=
κ
◦
◦
γ,
as
desired.
This
completes
the
proof
of
assertion
(iii).
Remark
5.10.1.
Observe
that
Theorem
5.10,
(iii),
may
be
interpreted
as
asserting
that
a
mono-theta
environment
may
be
“extracted”
from
the
tempered
Frobenioids
under
consideration
in
a
purely
category-theoretic
fashion.
In
particular,
by
coupling
this
observation
with
Corollary
2.18
[cf.
also
Remark
2.18.2],
we
conclude
that:
A
mono-theta
environment
may
be
“extracted”
naturally
from
both
the
tempered
Frobenioids
and
the
tempered
fundamental
groups
under
consid-
eration
in
a
purely
category/group-theoretic
fashion.
That
is
to
say,
a
mono-theta
environment
may
be
thought
of
as
a
sort
of
min-
imal
core
common
to
both
the
[tempered-]étale-theoretic
and
Frobenioid-theoretic
approaches
to
the
theta
function
[cf.
Remark
2.18.2].
Put
another
way,
from
the
point
of
view
of
the
general
theory
of
Frobenioids,
although
the
““étale-like”
[e.g.,
temperoid]”
and
“Frobenius-like”
portions
of
a
Frobenioid
are
fundamentally
alien
to
one
another
in
nature
[cf.
the
“fundamental
dichotomy”
discussed
in
[Mzk17],
Remark
3.1.3],
a
mono-theta
environment
serves
as
a
sort
of
bridge,
relative
to
the
theory
of
theta
function,
between
these
two
fundamentally
mutually
alien
aspects
of
the
structure
of
a
Frobenioid.
Remark
5.10.2.
The
structure
of
a
Frobenioid
may
be
thought
of
as
consisting
of
a
sort
of
extension
structure
of
the
base
category
by
various
line
bundles.
From
this
point
of
view:
The
theta
section
portion
of
a
mono-theta
environment
may
be
thought
as
a
sort
of
canonical
splitting
of
this
extension,
determined
by
the
theory
of
the
étale
theta
function
[cf.
the
discussion
of
canonicality/rigidity
in
the
Introduction].
This
point
of
view
is
reminiscent
[cf.
the
point
of
view
of
Remark
5.10.1]
of
the
no-
tion
of
a
“canonical
uniformizing
MF
∇
-object”
discussed
in
[Mzk1],
Introduction,
§1.3.
Remark
5.10.3.
One
key
feature
of
a
mono-theta
environment
is
the
inclusion
in
the
mono-theta
environment
of
the
“distinct
cyclotome”
μ
N
(B
N
)
∼
=
Ker(E
Π
N
tp
tp
Π
Y
)
[i.e.,
a
cyclotome
distinct
from
the
various
cyclotomes
associated
to
Π
Y
].
Here,
we
pause
to
observe
that:
This
“distinct
cyclotome”
may
be
thought
of
as
a
sort
of
“Frobenius
germ”
—
i.e.,
a
“germ”
or
“trace”
of
the
“Frobenius
structure”
of
the
tempered
Frobenioid
C
constituted
by
raising
elements
of
O
(−)
to
N
≥1
-powers.
THE
ÉTALE
THETA
FUNCTION
101
Indeed,
when
a
mono-theta
environment
is
not
considered
as
a
separate,
abstract
mathematical
structure,
but
rather
as
a
mathematical
structure
associated
to
the
tempered
Frobenioid
C,
the
operation
of
“raising
to
N
≥1
-powers”
elements
of
O
(−)
in
C
is
compatible
with
the
natural
multiplication
action
of
N
≥1
on
this
distinct
cyclotome.
Moreover,
these
actions
of
N
≥1
are
compatible
with
the
operation
of
forming
Kummer
classes
[e.g.,
passing
from
the
Frobenioid-theoretic
version
of
the
theta
function
to
its
Kummer
class,
the
“étale
theta
function”],
as
well
as
with
the
consideration
of
values
∈
K
×
of
functions
[e.g.,
the
theta
function
—
cf.
Proposition
1.4,
(iii)],
relative
to
the
reciprocity
map
on
elements
of
K
×
[cf.
[Mzk18],
Theorem
2.4].
On
the
other
hand,
it
is
important
to
note
that:
These
actions
of
N
≥1
only
make
sense
within
the
tempered
Frobenioid
C;
that
is
to
say,
they
do
not
give
rise
to
an
action
of
N
≥1
on
the
mono-theta
environment
[considered
as
a
separate,
abstract
mathematical
structure].
Indeed,
the
fact
that
one
does
not
obtain
a
natural
action
of
N
≥1
on
the
mono-theta
environment
may
be
understood,
for
instance,
by
observing
that
the
cyclotomic
rigidity
isomorphism
∼
(l
·
Δ
Θ
)
⊗
(Z/N
Z)
→
Δ
[μ
N
]
[cf.
Corollary
2.19,
(i);
Remark
2.19.3]
is
not
compatible
with
the
endomorphism
of
the
“distinct
cyclotome”
Δ
[μ
N
]
given
by
multiplication
by
M
∈
N
≥1
,
unless
M
≡
1
modulo
N
.
[Here,
it
is
useful
to
recall
that
there
is
no
natural
action
of
N
≥1
on
Π
tp
X
that
induces
the
multiplication
action
on
N
≥1
on
l
·
Δ
Θ
!]
Alternatively,
this
property
may
be
regarded
as
a
restatement
of
the
interpretation
[cf.
Remark
2.19.3]
of
the
cyclotomic
rigidity
of
a
mono-theta
environment
as
a
sort
of
“integral
structure”,
or
“basepoint”,
relative
to
the
action
of
N
≥1
.
Moreover,
we
remark
that
it
is
precisely
the
presence
of
this
rigidity
property
that
motivates
the
interpretation,
stated
above,
of
the
“distinct
cyclotome”
as
a
sort
of
“Frobenius
germ”
—
i.e.,
something
that
is
somewhat
less
than
the
“full
Frobenius
structure”
present
in
a
Frobenioid,
but
which
nevertheless
serves
as
a
sort
of
“trace”,
or
“partial,
but
essential
record”,
of
such
a
“full
Frobenius
structure”.
Remark
5.10.4.
(i)
Relative
to
the
theory
of
§2,
which
does
not
involve
Frobenioids,
the
signif-
icance
of
the
introduction
of
Frobenioids
may
be
understood
as
follows:
The
use
of
Frobenioids
allows
one
to
consider,
in
a
natural
way,
the
monoid
√
(Θ̈(
−1)
−1
·
Θ̈)
N
[cf.
Theorem
5.7]
of
powers
of
the
theta
function.
This
is
not
possible
for
“abstract
mono-theta
environments”,
for
numerous
reasons,
centering
around
the
preservation
of
cyclotomic
rigidity
[cf.
Remarks
2.19.2,
2.19.3,
5.10.3,
as
well
as
Remark
5.12.5
below].
One
way
to
understand
this
phenomenon
102
SHINICHI
MOCHIZUKI
is
to
think
of
the
various
discrete
monoids
[i.e.,
“copies
of
N”]
that
appear
in
the
structure
of
a
Frobenioid
as
imparting
a
rigidity
to
Frobenioids
that
takes
the
place
of
the
cyclotomic
rigidity
of
an
abstract
mono-theta
environment
[cf.
Remark
2.19.4].
From
this
point
of
view,
the
tempered
Frobenioid
C
involved
may
be
thought
of
as
furnishing
a
sort
of
“discrete
Tate-module-like
unraveling”
—
i.e.,
a
means
of
passing
from
“Q/Z(1)”
to
“N”
[as
opposed
to
Hom(Q/Z(1),
Q/Z(1))
=
Z]
—
of
the
various
abstract
mono-theta
environments
that
appear
in
the
theory
[cf.
the
notion
of
“Frobenius
germs”
discussed
in
Remark
5.10.3].
√
(ii)
If
one
is
only
interested
in
the
monoid
“(Θ̈(
−1)
−1
·
Θ̈)
N
”
of
(i),
then
it
might
appear
at
first
glance
that
instead
of
working
with
the
tempered
Frobenioid
C,
it
suffices
instead
to
work
with
the
associated
birationalization
C
birat
.
On
the
other
hand,
it
is
precisely
by
working
with
C
[instead
of
C
birat
],
i.e.,
by
working
with
ample
line
bundles,
that
one
may
access
the
“numerator”
and
“denominator”
of
a
meromorphic
function
as
separate,
independent
entities.
Moreover,
it
is
this
aspect
of
C
that
is
of
crucial
importance
in
relating
the
Frobenioid-theoretic
aspects
of
(i)
to
an
abstract
mono-theta
[i.e.,
as
opposed
to
bi-theta,
which
would
force
one
to
sacrifice
discrete
rigidity
—
cf.
Remark
2.16.1]
environment.
(iii)
In
addition
to
the
significance
of
the
use
of
Frobenioids
as
discussed
in
(i),
more
generally
the
introduction
of
Frobenioids
is
useful
in
that,
unlike
tempered
fundamental
groups,
Frobenioids
tend
to
be
compatible
with
“Frobenius-like”
struc-
tures
[e.g.,
structures
that
involve
some
sort
of
linear
ordering]
—
cf.
the
discussion
of
[Mzk17],
Remark
3.1.3.
By
contrast,
by
thinking
of
an
abstract
mono-theta
envi-
ronment
as
arising
from
the
tempered
fundamental
group
[cf.
Corollary
2.18],
one
may
work
with
such
mono-theta
environments
in
situations
in
which
it
is
important
to
be
free
of
the
constraints
arising
from
such
“Frobenius-like”,
“order-conscious”
structures.
Before
proceeding,
we
review
a
“well-known”
result
in
category
theory.
Lemma
5.11.
(Non-category-theoreticity
of
Particular
Morphisms
in
an
Abstract
Equivalence
Class)
Let
E
be
a
category;
G,
H,
I
∈
Ob(E)
distinct
objects
of
E;
f
:
G
→
H,
g
:
G
→
G
morphisms
of
E;
α
G
∈
Aut
E
(G),
α
H
∈
Aut
E
(H),
α
I
∈
Aut
E
(I)
automorphisms.
Then
there
exists
a
self-equivalence
∼
Ξ
:
E
→
E
that
induces
the
identity
on
Ob(E)
and
is
isomorphic
to
the
identity
self-equivalence
via
an
isomorphism
that
maps
G
→
α
G
∈
Aut
E
(G);
H
→
α
H
∈
Aut
E
(H);
I
→
α
I
∈
Aut
E
(I);
J
→
id
J
∈
Aut
E
(J
)
—
where
id
J
denotes
the
identity
morphism
J
→
J
—
for
all
J
∈
Ob(E)
such
that
−1
J
=
G,
H,
I.
In
particular,
Ξ
maps
f
→
α
H
◦
f
◦
α
−1
G
,
g
→
α
G
◦
g
◦
α
G
.
Proof.
First,
let
us
observe
that
by
composing
the
self-equivalences
obtained
by
applying
Lemma
5.11
with
two
of
the
three
automorphisms
α
G
,
α
H
,
α
I
taken
to
be
THE
ÉTALE
THETA
FUNCTION
103
the
identity,
we
may
assume
without
loss
of
generality
that
α
H
,
α
I
are
the
respective
identity
automorphisms
of
H,
I.
Now
[one
verifies
immediately
that]
one
may
define
an
equivalence
of
categories
[that
is
isomorphic
to
the
identity
self-equivalence]
∼
Ξ:
E
→E
which
restricts
to
the
identity
on
Ob(E)
and
maps
j
→
j
◦
(α
G
)
−1
;
j
→
α
G
◦
j
;
j
→
α
G
◦
j
◦
α
−1
G
;
j
→
j
for
j
∈
Hom
E
(G,
J),
j
∈
Hom
E
(J,
G),
j
∈
Hom
E
(G,
G),
j
∈
Hom
E
(J,
J
),
where
G
=
J,
J
∈
Ob(E).
Thus,
Ξ
satisfies
the
properties
asserted
in
the
statement
of
Lemma
5.11.
Finally,
we
apply
the
“general
nonsense
category
theory”
of
Lemma
5.11
to
explain
certain
aspects
of
the
motivation
that
underlies
the
theory
of
mono-theta
environments.
Corollary
5.12.
(Constant
Multiple
Indeterminacy
of
Systems)
In
the
notation
of
Theorem
5.10,
assume
further
that
N
≥
1
is
an
integer
such
that
N
def
divides
N
,
but
M
=
N
/N
=
1;
s
N
,
s
N
:
A
N
→
B
N
an
N
-th
root
of
a
right
fraction-pair
of
an
l-th
root
of
the
theta
function
Θ̈
such
that
there
exists
a
pair
of
commutative
diagrams
s
N
A
N
−→
⏐
⏐
α
N,N
A
N
s
N
−→
s
B
N
⏐
⏐
β
N,N
N
A
N
−→
⏐
⏐
α
N,N
B
N
A
N
s
N
−→
B
N
⏐
⏐
β
N,N
B
N
—
where
α
N,N
(respectively,
β
N,N
)
is
an
isometry
of
Frobenius
degree
M
=
1;
α
N,N
is
of
base-Frobenius
type
[cf.
Remark
4.3.2].
Then:
(i)
The
isomorphism
classes
of
A
N
,
B
N
,
and
B
N
are
distinct.
(ii)
There
exists
a
linear
morphism
ι
:
B
N
→
B
N
[cf.
the
proof
of
Proposition
5.5].
(iii)
Let
ζ
:
B
N
→
B
N
×
1/N
be
either
β
N,N
or
ι.
Write
(O
K
)
|
A
⊆
O
×
(A
N
)
for
the
subgroup
induced
by
×
1/N
×
∗
)
⊆
O
×
(B
N
)
[cf.
Lemma
5.8]
via
s
N
or
s
N
;
(O
K
)
⊆
O
×
(B
N
)
for
the
(O
K
subgroup
×
1/N
M/deg
Fr
(ζ)
)
)
((O
K
×
∗
[cf.
Lemma
5.8].
[Thus,
we
have
a
natural
μ
N
-outer
action
of
(O
K
)
/μ
N
(B
N
)
on
Aut
C
(B
N
)
that
is
compatible,
relative
to
ζ,
with
the
natural
μ
N
-outer
action
of
104
SHINICHI
MOCHIZUKI
×
1/N
×
1/N
×
1/N
(O
K
)
/μ
N
(B
N
)
on
Aut
C
(B
N
).]
Then
for
any
κ
A
∈
(O
K
)
|
A
,
κ
B
∈
(O
K
)
,
×
∗
)
,
there
exists
a
self-equivalence
κ
∈
(O
K
∼
Ξ:
C
→C
that
is
isomorphic
to
the
identity
self-equivalence
via
an
isomorphism
that
maps
A
N
→
κ
−1
A
∈
Aut
C
(A
N
);
B
N
→
κ
−1
B
∈
Aut
C
(B
N
);
B
N
→
(κ
)
−1
∈
Aut
C
(B
N
)
and
all
other
objects
of
C
to
the
corresponding
identity
automorphism.
In
particular,
Ξ
maps
s
N
→
κ
−1
B
◦
s
N
◦
κ
A
;
and
s
N
→
κ
−1
B
◦
s
N
◦
κ
A
;
ζ
→
κ
−1
B
◦
ζ
◦
κ
;
−1
trv
Im(s
trv
N
)
(⊆
Aut
C
(A
N
))
→
κ
A
·
Im(s
N
)
·
κ
A
⊆
Aut
C
(A
N
);
-gp
)
·
κ
B
⊆
Aut
C
(B
N
);
Im(s
N
-gp
)
(⊆
Aut
C
(B
N
))
→
κ
−1
B
·
Im(s
N
-gp
)
·
κ
B
⊆
Aut
C
(B
N
)
Im(s
N
-gp
)
(⊆
Aut
C
(B
N
))
→
κ
−1
B
·
Im(s
N
-gp
”,
“s
N
-gp
”
are
as
in
the
discussion
preceding
Lemma
5.8.
—
where
“s
trv
N
”,
“s
N
Proof.
First,
we
consider
assertions
(i),
(ii).
Since
β
N,N
is
an
isometry
of
Frobenius
bs
bs
degree
M
,
it
follows
that
the
pull-back
via
the
morphism
B
N
→
B
N
of
D
of
the
line
bundle
that
determines
the
object
B
N
is
isomorphic
to
the
M
-th
tensor
power
of
the
line
bundle
that
determines
the
object
B
N
.
Moreover,
it
follows
immediately
from
the
discussion
of
line
bundles
in
§1
[cf.
the
discussion
preceding
Proposition
1.1]
that
all
positive
tensor
powers
of
these
line
bundles
are
nontrivial,
and
that
the
isomorphism
classes
of
these
line
bundles
are
preserved
by
arbitrary
bs
automorphisms
of
B
N
.
In
particular,
we
conclude
immediately
that
B
N
,
B
N
are
non-isomorphic
both
to
one
another
and
to
the
Frobenius-trivial
object
[i.e.,
object
defined
by
a
trivial
line
bundle]
A
N
.
Moreover,
by
multiplying
by
the
(M
−
1)-th
tensor
power
of
the
section
of
line
bundles
that
determines
either
of
the
morphisms
s
N
,
s
N
:
A
N
→
B
N
[cf.
the
discussion
of
Remark
5.12.5,
(iii),
below],
we
obtain
a
linear
morphism
ι
:
B
N
→
B
N
.
This
completes
the
proof
of
assertions
(i),
(ii).
Finally,
in
light
of
assertion
(i),
assertion
(iii)
follows
immediately
from
Lemma
5.11.
Remark
5.12.1.
Let
E
be
a
category.
Then
for
any
E
∈
Ob(E),
Aut
E
(E)
has
a
natural
group
structure.
Indeed,
this
group
structure
is
precisely
the
group
structure
that
allows
one,
for
instance,
to
represent
the
group
structure
of
a
tem-
pered
topological
group
category-theoretically
via
temperoids
or
to
represent
[cf.
THE
ÉTALE
THETA
FUNCTION
105
Theorem
5.10,
(iii)]
the
group
structure
portion
of
a
mono-theta
environment
cate-
gory-theoretically
via
tempered
Frobenioids.
On
the
other
hand,
in
both
of
these
cases,
one
is,
in
fact,
not
just
interested
in
the
group
structure,
but
rather
in
the
topological
group
structure
of
the
various
objects
under
consideration.
From
this
point
of
view:
Temperoids
[or,
for
that
matter,
Galois
categories]
allow
one
to
represent
the
topological
group
structure
under
consideration
via
the
use
of
numer-
ous
objects
corresponding
to
the
various
open
subgroups
of
the
topologi-
cal
group
under
consideration,
as
opposed
to
the
use
of
a
“single
universal
covering
object”,
whose
automorphism
group
allows
one
to
represent
the
entire
group
under
consideration
via
a
single
object,
but
only
at
the
ex-
pense
of
sacrificing
the
additional
data
that
constitutes
the
topology
of
the
topological
group
under
consideration.
This
approach
of
using
“numerous
objects
corresponding
to
the
various
open
sub-
groups”
carries
over,
in
effect,
to
the
theory
of
tempered
Frobenioids,
since
such
tempered
Frobenioids
typically
appear
over
base
categories
given
by
[the
subcate-
gory
of
connected
objects
of]
a
temperoid.
Remark
5.12.2.
In
general,
if
one
tries
to
consider
systems
[e.g.,
projective
systems,
such
as
“universal
coverings”]
in
the
context
of
the
“numerous
objects
approach”
of
Remark
5.12.1,
then
one
must
contend
with
the
following
problem:
Suppose
that
to
each
object
E
in
some
given
given
collection
I
of
isomorphism
classes
of
objects
of
E,
one
associates
certain
data
E
→
D
E
[such
as
the
group
Aut
E
(E),
or
some
category-theoretically
determined
subquotient
of
Aut
E
(E)],
which
one
may
think
of
as
a
functor
on
the
full
subcategory
of
E
consisting
of
objects
whose
isomorphism
class
belongs
to
I.
Then:
One
must
contend
with
the
fact
there
is
no
natural,
“category-theore-
tic”
choice
[cf.
Lemma
5.11]
of
a
particular
morphism
ζ
:
E
→
F
,
among
the
various
composites
α
F
◦
ζ
◦
α
E
—
where
α
E
∈
Aut
E
(E),
α
F
∈
Aut
E
(F
)
—
for
the
task
of
relating
D
E
to
D
F
.
In
particular:
One
necessary
condition
for
the
data
constituted
by
the
functor
E
→
D
E
to
form
a
coherent
system
is
the
condition
that
the
data
D
E
be
invariant
with
respect
to
the
various
automorphisms
induced
by
the
various
Aut
E
(E)
—
i.e.,
that
Aut
E
(E)
act
as
the
identity
on
D
E
.
106
SHINICHI
MOCHIZUKI
One
“classical”
example
of
this
phenomenon
is
the
category-theoretic
reconstruction
of
a
tempered
topological
group
from
its
associated
temperoid
[cf.
[Mzk14],
Propo-
sition
3.2;
the
even
more
classical
case
of
Galois
categories],
where
one
is
obliged
to
work
with
topological
groups
up
to
inner
automorphism
[cf.
also
Remark
5.12.8
below].
Remark
5.12.3.
Now
we
return
to
our
discussion
of
Frobenioid-theoretic
mono-
theta
environments,
in
the
context
of
Theorem
5.10,
(iii).
If,
instead
of
working
with
“finite”
mono-theta
environments,
one
attempts
to
work
with
the
projective
system
of
mono-theta
environments
determined
by
letting
N
vary
[cf.
Remark
2.16.1],
then
one
must
contend
with
the
“constant
multiple
indeterminacy”
of
Corollary
5.12,
(iii),
relative
to
κ
,
of
ζ
=
β
N,N
[i.e.,
where
we
note
that
β
N,N
may
be
thought
of
as
a
typical
morphism
appearing
in
this
projective
system].
In
particu-
lar,
the
existence
of
this
indeterminacy
implies
that,
in
order
to
obtain
the
analogue
of
Theorem
5.10,
(iii)
—
i.e.,
to
describe,
in
a
category-theoretic
fashion,
the
rela-
tionship
between
a
single
abstract,
static
external
projective
system
of
mono-theta
environments
and
the
projective
system
of
mono-theta
environments
constructed
inside
a
tempered
Frobenioid
—
one
must
work
with
mono-theta
environments
up
to
the
indeterminacies
arising
from
the
μ
N
-outer
action
of
∼
×
1/N
×
1/N
×
Aut
C
(B
N
)
⊇
O
×
(B
N
)
⊇
(O
K
)
(O
K
)
/μ
N
(B
N
)
→
O
K
[cf.
Lemma
5.8].
That
is
to
say,
one
must
assume
that
one
only
knows
the
theta
section
portion
of
a
mono-theta
environment
[cf.
Definition
2.13,
(ii),
(c)]
up
to
a
constant
multiple.
Put
another
way,
one
is
forced
to
sacrifice
the
constant
multiple
rigidity
of
Corollary
2.19,
(iii).
Moreover,
we
observe
in
passing
that
attempting
to
avoid
sacrificing
“constant
multiple
rigidity”
by
working
with
bi-theta
environments
[cf.
the
discussion
preceding
Lemma
5.8]
does
not
serve
to
remedy
this
situation,
since
this
forces
one
to
sacrifice
“discrete
rigidity”
[cf.
Remark
2.16.1].
Thus,
in
summary:
A
[finite,
not
profinite!]
mono-theta
enviroment
serves
in
effect
to
max-
imize
the
rigidity,
i.e.,
to
minimize
the
indeterminacy,
of
the
[l-th
roots
of
the]
theta
function
that
one
works
with
in
the
following
three
crucial
respects:
(a)
cyclotomic
rigidity
[cf.
Corollary
2.19,
(i);
Remark
2.19.4];
(b)
discrete
rigidity
[cf.
Corollary
2.19,
(ii);
Remarks
2.16.1,
2.19.4];
(c)
constant
multiple
rigidity
[cf.
Corollary
2.19,
(iii);
the
discus-
sion
of
the
present
Remark
5.12.3;
Remark
5.12.5
below]
—
all
in
a
fashion
that
is
compatible
with
the
category-theoretic
represen-
tation
of
the
topology
of
the
tempered
fundamental
group
discussed
in
Remark
5.12.1.
This
“extraordinary
rigidity”
of
a
mono-theta
environment,
along
with
the
“bridging
aspect”
discussed
in
Remarks
2.18.2,
5.10.1,
5.10.2,
5.10.3,
were,
from
the
point
of
THE
ÉTALE
THETA
FUNCTION
107
view
of
the
author,
the
main
motivating
reasons
for
the
introduction
of
the
notion
of
a
mono-theta
environment.
Remark
5.12.4.
With
regard
to
the
projective
systems
of
mono-theta
environ-
ments
discussed
in
Remark
5.12.3,
if,
instead
of
trying
to
relate
a
single
abstract
such
system
to
a
Frobenioid-theoretic
system,
one
instead
takes
the
approach
of
relating,
at
each
finite
step
[i.e.,
at
each
constituent
object
of
the
system],
an
ab-
stract
mono-theta
environment
to
a
Frobenioid-theoretic
mono-theta
environment
via
Theorem
5.10,
(iii)
—
hence,
in
particular,
taking
into
account
the
“Aut
C
(B
N
)-
orbit”
indeterminacies
that
occur
at
each
step
when
one
considers
such
relationships
—
then
one
can
indeed
establish
a
category-theoretic
relationship
between
projec-
tive
systems
of
mono-theta
environments
external
and
internal
to
the
Frobenioid
C
without
sacrificing
the
cyclotomic,
discrete,
or
constant
multiple
rigidity
properties
discussed
in
Remark
5.12.3.
Put
another
way,
the
difference
between
“attempting
to
relate
whole
systems
at
once
as
in
Remark
5.12.3”
and
“applying
Theorem
5.10,
(iii),
at
each
finite
step”
may
be
thought
of
as
a
matter
of
“how
one
arranges
one’s
parentheses”,
that
is
to
say,
as
the
difference
between
.
.
.
−→
external
N
−→
⏐
⏐
external
N
−→
.
.
.
.
.
.
−→
internal
N
−→
internal
N
−→
.
.
.
[i.e.,
the
approach
discussed
in
Remark
5.12.3,
which
forces
one
to
sacrifice
constant
multiple
rigidity]
and
⎛
...
⎜
−→
⎜
⎝
external
N
⏐
⏐
internal
N
⎞
⎛
⎟
⎟
⎠
⎜
−→
⎜
⎝
external
N
⏐
⏐
⎞
⎟
⎟
⎠
−→
.
.
.
internal
N
[i.e.,
“applying
Theorem
5.10,
(iii),
at
each
finite
step”
—
cf.
the
projective
systems
discussed
in
Corollary
2.19,
(iii)].
Remark
5.12.5.
(i)
Suppose
that
instead
of
working
with
first
powers
of
l-th
roots
of
the
theta
function,
one
tries
instead
to
consider
an
M
-th
power
version
[where
M
>
1
is
an
integer]
of
the
mono-theta
environment.
As
discussed
in
Remark
2.19.2,
if
one
takes
the
most
naive
approach
to
doing
this,
then
one
must
sacrifice
cyclotomic
rigidity
in
the
sense
that
instead
of
obtaining
cyclotomic
rigidity
for
“μ
N
”
[where,
say,
N
is
divisible
by
M
],
one
obtains
cyclotomic
rigidity
only
for
the
submodule
“M
·
μ
N
⊆
μ
N
”.
Moreover,
the
resulting
structures
end
up
being
intrisically
indistinguishable
[cf.
Remark
2.19.3]
from
their
first
power
counterparts,
where
108
SHINICHI
MOCHIZUKI
“N
”
is
replaced
by
“N/M
”.
In
particular,
in
order
to
consider
an
“M
-th
power
version”
of
the
mono-theta
environment
in
a
meaningful
fashion,
one
is
led
to
consider,
in
addition,
[the
cyclotomic
rigidity
isomorphism
arising
from]
some
“first
power
version”,
together
with
data
exhibiting
the
“M
-th
power
version”
as
the
M
-th
power
of
the
“first
power
version”.
(ii)
Thus,
one
is
led,
for
instance,
to
consider
systems
consisting
of
the
following
data:
(a)
an
M
-th
power
version
of
the
mono-theta
environment;
(b)
a
first
power
version
[i.e.,
the
usual
version]
of
the
mono-theta
environment;
(c)
data
relating
these
two
versions
[i.e.,
to
the
effect
that
the
“M
-th
power
version”
is
indeed
ob-
tained
as
the
M
-th
power
of
the
“first
power
version”].
Here,
(c)
amounts,
in
effect,
to
the
consideration
of
morphisms
such
as
the
morphisms
ι
or
β
N,N
of
Corollary
5.12.
In
particular,
the
phenomenon
discussed
in
Remark
5.12.2
manifests
itself,
i.e.,
one
must
contend
with
the
constant
multiple
indeterminacy
discussed
in
Corol-
lary
5.12,
(iii).
(iii)
One
approach
to
attempting
to
avoid
the
constant
multiple
indeterminacy
that
arose
in
(ii)
[i.e.,
avoid
the
use
of
“ι”,
“β
N,N
”]
is
to
consider
collections
of
data
consisting
of
the
following:
(a)
an
M
-th
power
version
of
the
mono-theta
environ-
ment;
(b)
the
Frobenioid-theoretic
cyclotomic
rigidity
isomorphism
of
Proposition
5.5.
At
first
glance,
this
allows
one
to
avoid
the
“constant
multiple
indetermi-
nacy”
that
arises
from
working
with
“systems
of
objects”
[cf.
Remark
5.12.2],
by
restricting
the
data
to
data
associated
to
a
single
object
of
the
Frobenioid.
Closer
inspection,
however,
reveals
a
more
complicated
picture.
Indeed,
recall
from
the
proof
of
Proposition
5.5
that
the
isomorphism
of
(b)
was
obtained
by
transporting,
via
linear
morphisms,
the
cyclotomic
rigidity
isomorphism
obtained
from
a
first
power
version
of
the
mono-theta
environment.
Moreover,
let
us
observe
that
this
transportation
operation
is
performed
on
the
data
Aut
D
((−)
bs
),
μ
N
(−)
[or,
more
generally,
O
×
(−)]
of,
say,
B
N
,
B
N
[cf.
the
proof
of
Proposition
5.5]
—
i.e.,
data
which
constitutes
a
sort
of
“semi-simplification”
of
the
“extension
structure”
of
a
Frobenioid
discussed
in
Remark
5.10.2.
On
the
other
hand,
there
is
no
evident
way
to
extend
such
a
transportation
operation
so
as
to
apply
to
the
“extension
structure”
[cf.
Remark
5.10.2]
of
the
exact
sequence
1
→
O
×
(−)
→
Aut
C
((−))
→
Aut
D
((−)
bs
)
→
1
[for
objects
“(−)”,
such
as
B
N
,
B
N
,
which
are
Aut-ample
—
cf.
the
discussion
preceding
Lemma
5.8].
Put
another
way,
to
perform
this
transportation
operation
requires
one
to
work
up
to
an
indeterminacy
with
respect
to
[so
to
speak
“unipotent
upper-triangular”]
“shifting
automorphisms”
arising
from
cocycles
of
Aut
D
((−)
bs
)
with
coefficients
in
O
×
(−)
(⊇
μ
N
(−))
[cf.
the
situation
of
Proposition
2.14,
(ii)],
×
which
includes,
in
particular,
a
constant
[i.e.,
O
K
-]
multiple
indeterminacy
[cf.
the
∼
×
1/N
×
)
/μ
N
(B
N
)
→
O
K
in
collection
of
cocycles
determined
by
the
image
of
(O
K
1
bs
H
(Aut
D
(B
N
),
μ
N
(B
N
)!].
More
generally,
by
considering
Kummer
classes
of
log-
meromorphic
functions
[such
as
the
étale
theta
function!
—
cf.
the
proof
of
Corol-
lary
5.12],
it
follows
that
this
“shifting
automorphism
indeterminacy”
implies,
in
particular,
an
indeterminacy
which
is,
in
effect,
a
sort
of
Galois/Kummer-theoretic
translation
of
working
with
line
bundles
birationally.
Note,
moreover,
that
the
THE
ÉTALE
THETA
FUNCTION
109
cyclotomic
rigidity
isomorphism
[cf.
Corollary
2.19,
(i)]
arising
from
the
specific
splitting
determined
by
the
theta
section
is
not
preserved
by
[so
to
speak
“unipo-
tent
upper-triangular”]
“shifting
automorphisms”.
Thus,
in
summary:
The
operation
of
relating
the
isomorphism
of
(b)
to
the
“M
-th
power
version”
of
such
a
cyclotomic
rigidity
isomorphism
arising
from
(a)
—
i.e.,
exhibiting
the
“M
-th
power
version”
of
(a)
as
the
M
-th
power
of
the
isomorphism
of
(b)
—
may
only
be
performed
after
passing
to
the
“semi-
simplified
data”
Aut
D
((−)
bs
),
μ
N
(−).
On
the
other
hand,
applying
the
operation
of
“multiplication
by
M
”
to
the
portion
of
the
“semi-simplified
data”
constituted
by
μ
N
(−)
amounts
precisely
to
the
situa-
tion
discussed
in
Remark
2.19.3
[cf.
also
(i)
above;
Remark
5.10.3].
That
is
to
say,
this
causes
problems
arising
from
the
intrinsic
indistinguishability
of
the
domain
and
codomain
copies
of
“μ
N
(−)”
that
occur
when
one
multiplies
by
M
.
On
the
other
hand,
to
attempt
to
assign
“distinct
labels”
to
these
domain
and
codomain
copies
of
“μ
N
(−)”
amounts,
in
essence,
to
working
with
non-isomorphic
tensor
powers
of
some
line
bundle,
i.e.,
to
returning,
in
effect,
to
the
situation
discussed
in
(ii)
above.
(iv)
Finally,
we
note
that
another
aspect
of
the
lack
of
canonical
splittings
of
the
exact
sequence
1
→
O
×
(−)
→
Aut
C
((−))
→
Aut
D
((−)
bs
)
→
1
of
(iii)
is
that,
relative
to
the
automorphism
indeterminacies
discussed
in
Remark
5.12.2,
there
is
no
natural
way
to
achieve
a
situation
in
which
the
inner
automorphism
indeterminacies
of
the
tempered
fundamental
group
[i.e.,
“Aut
D
((−)
bs
)”]
remain,
but
the
constant
multiple
indeterminacies
[i.e.,
“O
×
(−)”]
are
eliminated.
Remark
5.12.6.
At
this
point,
it
is
useful
to
reflect
[cf.
Remark
2.19.4]
on
the
significance
of
rigidifying
the
“constant
multiple
indeterminacy”
and
“shifting
automorphism
indeterminacy”
of
Remarks
5.12.3,
5.12.5.
To
this
end,
we
observe
that
these
types
of
indeterminacy
are
essentially
multiplicative
notions
[cf.
the
cases
discussed
in
Remark
2.19.4].
Thus,
to
work
“modulo
these
sorts
of
indeterminacy”
can
only
be
done
at
the
cost
of
sacrificing
the
additive
structures
[cf.
Remark
2.19.4]
implicit
in
the
ring/scheme-theoretic
origins
of
the
various
objects
under
consideration.
Another
important
observation
in
this
context
is
that
the
theory
of
Kummer
classes
depends
essentially
on
the
extension
structure
of
a
Frobenioid,
i.e.,
breaks
down
completely
if
one
“passes
to
semi-simplifications”,
as
in
Remark
5.12.5,
(iii).
Remark
5.12.7.
(i)
One
way
to
understand
the
“constant
multiple
indeterminacy”
phenomena
observed
in
Remarks
5.12.3,
5.12.5
is
as
a
manifestation
of
the
nontriviality
of
the
extension
structure
of
a
Frobenioid
discussed
in
Remark
5.10.2.
(ii)
Note
that
one
way
to
attempt
to
avoid
this
“nontrivial
extension
structure
of
a
Frobenioid”
[which
arises
essentially
from
working
with
nontrivial
line
bundles]
110
SHINICHI
MOCHIZUKI
is
to
try
to
work
strictly
with
rational
functions
[such
as
the
theta
function],
i.e.,
to
work
with
the
birationalizations
[i.e.,
in
which
“all
line
bundles
becomes
trivial”]
of
the
Frobenioid
“C”
of
Theorems
5.6,
5.10;
Corollary
5.12.
On
the
other
hand,
at
the
Galois/Kummer-theoretic
level,
this
amounts
to
working
strictly
with
étale
theta
functions
[i.e.,
without
theta
environments],
hence
gives
rise
to
the
same
“nondis-
creteness
phenomena”
as
those
that
appeared
in
the
case
of
bi-theta
environments
[cf.
Remarks
2.16.1;
5.10.4,
(ii)].
Remark
5.12.8.
Of
course,
the
automorphism
indeterminacy
with
regard
to
individual
objects
of
a
category
discussed
in
Remark
5.12.2
also
applies
to
the
var-
ious
base
categories
—
i.e.,
[essentially]
connected
temperoids
—
of
the
[tempered]
Frobenioids
that
appear
in
the
above
discussion.
On
the
other
hand,
one
verifies
immediately
that
the
various
objects
that
we
construct
out
of
the
associated
tem-
pered
fundamental
groups
are
invariant
with
respect
to
inner
automorphisms
in
a
fashion
that
is
compatible
with
the
topology
of
the
tempered
fundamental
group
[cf.
Remark
5.12.1].
For
instance,
the
construction
of
the
tempered
fundamental
group
of
a
connected
temperoid
involves
projective
systems
that
satisfy
the
analogue
[cf.
Remark
2.15.2]
of
the
discrete
rigidity
property
[cf.
Corollary
2.19,
(ii)]
of
a
mono-
theta
environment.
Moreover,
the
étale
theta
function
is
a
collection
of
cohomology
classes
in
the
[continuous]
cohomology
of
the
tempered
fundamental
group,
hence
is
invariant
with
respect
to
inner
automorphisms.
Remark
5.12.9.
Perhaps
a
sort
of
“unifying
principle”
underlying
the
“constant
multiple
indeterminacy”
phenomena
observed
in
Remarks
5.12.3,
5.12.5,
on
the
one
hand,
and
the
discrete
rigidity
discussed
in
Remark
2.16.1,
on
the
other,
may
be
expressed
in
the
following
fashion:
Although
at
first
glance,
two
[or
many]
pieces
of
data
may
appear
to
be
likely
to
yield
“more
information”
than
“one”
piece
of
data,
in
fact,
the
more
pieces
of
data
that
one
considers
the
greater
the
indeterminacies
are
that
arise
in
describing
the
internal
relations
between
these
pieces
of
data.
Moreover,
these
greater
indeterminacies
may
[as
in
the
case
of
systems
of
objects
of
a
Frobenioid
in
Remarks
5.12.3,
5.12.5,
or
bi-theta
environments
in
Remark
2.16.1]
ultimately
result
in
“less
information”
than
the
information
resulting
from
a
single
piece
of
data
[cf.
the
discussion
in
the
Introduction].
THE
ÉTALE
THETA
FUNCTION
111
Bibliography
[André]
Y.
André,
On
a
Geometric
Description
of
Gal(Q
p
/Q
p
)
and
a
p-adic
Avatar
of
,
Duke
Math.
J.
119
(2003),
pp.
1-39.
GT
[FC]
G.
Faltings
and
C.-L.
Chai,
Degenerations
of
Abelian
Varieties,
Springer
Verlag
(1990).
[LynSch]
R.
C.
Lyndon
and
P.
E.
Schupp,
Combinatorial
Group
Theory,
Ergeb.
Math.
Grenzgeb.
89,
Springer
(1977).
[Mzk1]
S.
Mochizuki,
Foundations
of
p-adic
Teichmüller
Theory,
AMS/IP
Studies
in
Advanced
Mathematics
11,
American
Mathematical
Society/International
Press
(1999).
[Mzk2]
S.
Mochizuki,
The
Absolute
Anabelian
Geometry
of
Hyperbolic
Curves,
Galois
Theory
and
Modular
Forms,
Kluwer
Academic
Publishers
(2003),
pp.
77-122.
[Mzk3]
S.
Mochizuki,
The
Absolute
Anabelian
Geometry
of
Canonical
Curves,
Kazuya
Kato’s
fiftieth
birthday,
Doc.
Math.
2003,
Extra
Vol.,
pp.
609-640.
[Mzk4]
S.
Mochizuki,
A
Survey
of
the
Hodge-Arakelov
Theory
of
Elliptic
Curves
I,
Arithmetic
Fundamental
Groups
and
Noncommutative
Algebra,
Proceedings
of
Symposia
in
Pure
Mathematics
70,
American
Mathematical
Society
(2002),
pp.
533-569.
[Mzk5]
S.
Mochizuki,
A
Survey
of
the
Hodge-Arakelov
Theory
of
Elliptic
Curves
II,
Algebraic
Geometry
2000,
Azumino,
Adv.
Stud.
Pure
Math.
36,
Math.
Soc.
Japan
(2002),
pp.
81-114.
[Mzk6]
S.
Mochizuki,
The
Hodge-Arakelov
Theory
of
Elliptic
Curves:
Global
Discretiza-
tion
of
Local
Hodge
Theories,
RIMS
Preprint
Nos.
1255,
1256
(October
1999).
[Mzk7]
S.
Mochizuki,
The
Scheme-Theoretic
Theta
Convolution,
RIMS
Preprint
No.
1257
(October
1999).
[Mzk8]
S.
Mochizuki,
Connections
and
Related
Integral
Structures
on
the
Universal
Extension
of
an
Elliptic
Curve,
RIMS
Preprint
1279
(May
2000).
[Mzk9]
S.
Mochizuki,
The
Galois-Theoretic
Kodaira-Spencer
Morphism
of
an
Elliptic
Curve,
RIMS
Preprint
No.
1287
(July
2000).
[Mzk10]
S.
Mochizuki,
The
Hodge-Arakelov
Theory
of
Elliptic
Curves
in
Positive
Char-
acteristic,
RIMS
Preprint
No.
1298
(October
2000).
[Mzk11]
S.
Mochizuki,
The
Local
Pro-p
Anabelian
Geometry
of
Curves,
Invent.
Math.
138
(1999),
pp.
319-423.
[Mzk12]
S.
Mochizuki,
The
Geometry
of
Anabelioids,
Publ.
Res.
Inst.
Math.
Sci.
40
(2004),
pp.
819-881.
[Mzk13]
S.
Mochizuki,
Galois
Sections
in
Absolute
Anabelian
Geometry,
Nagoya
Math.
J.
179
(2005),
pp.
17-45.
112
SHINICHI
MOCHIZUKI
[Mzk14]
S.
Mochizuki,
Semi-graphs
of
Anabelioids,
Publ.
Res.
Inst.
Math.
Sci.
42
(2006),
pp.
221-322.
[Mzk15]
S.
Mochizuki,
A
combinatorial
version
of
the
Grothendieck
conjecture,
Tohoku
Math.
J.
59
(2007),
pp.
455-479.
[Mzk16]
S.
Mochizuki,
Absolute
anabelian
cuspidalizations
of
proper
hyperbolic
curves,
J.
Math.
Kyoto
Univ.
47
(2007),
pp.
451-539.
[Mzk17]
S.
Mochizuki,
The
Geometry
of
Frobenioids
I:
The
General
Theory,
Kyushu
J.
Math.
62
(2008),
pp.
293-400.
[Mzk18]
S.
Mochizuki,
The
Geometry
of
Frobenioids
II:
Poly-Frobenioids,
Kyushu
J.
Math.
62
(2008),
pp.
401-460.
[Mumf]
D.
Mumford,
An
Analytic
Construction
of
Degenerating
Abelian
Varieties
over
Complete
Rings,
Appendix
to
[FC].
Research
Institute
for
Mathematical
Sciences
Kyoto
University
Kyoto
606-8502,
Japan
Fax:
075-753-7276
motizuki@kurims.kyoto-u.ac.jp